We use dynamic programming to compute the domination number of the Cartesian product of two directed paths, \( \overrightarrow{P}_m \) and \( \overrightarrow{P}_n \), for \( m \leq 25 \) and all \( n \). This suggests that the domination number for \(\min(m,n) \geq 4\) is \( \left\lfloor \frac{(m+1)(n+1)}{3} \right\rfloor – 1 \), which we then confirm by showing that this is both an upper and a lower bound on the domination number.
In 1975, Erdős proposed the problem of determining the maximal number of edges in a graph on \( n \) vertices that contains no triangles or squares. In this paper, we consider a generalized version of the problem, i.e., what is the maximum size, \( ex(n; t) \), of a graph of order \( n \) and girth at least \( t+1 \) (containing no cycles of length less than \( t+1 \)). The set of those extremal \( C_t \)-free graphs is denoted by \( EX(n; t) \). We consider the problem on special types of graphs, such as pseudotrees, cacti, graphs lying in a square grid, Halin, generalized Halin, and planar graphs. We give the extremal cases, some constructions, and we use these results to obtain general lower bounds for the problem in the general case.
This paper develops the polyhedral approach to integer partitions. We consider the set of partitions of an integer \( n \) as a polytope \( P_n \subset \mathbb{R}^n \). Vertices of \( P_n \) form the class of partitions that provide the first basis for the whole set of partitions of \( n \). Moreover, we show that there exists a subclass of vertices, from which all others can be generated with the use of two combinatorial operations. The calculation demonstrates a considerable decrease in the cardinality of these classes of basic partitions as \( n \) grows. We focus on the vertex enumeration problem for \( P_n \). We prove that vertices of all partition polytopes form a partition ideal of the Andrews partition lattice. This allows us to construct vertices of \( P_n \) by a lifting method, which requires examining only certain partitions of \( n \). A criterion of whether a given partition is a convex combination of two others connects vertices with knapsack partitions, sum-free sets, Sidon sets, and Sidon multisets introduced in the paper. All but a few non-vertices for small \( n \)’s were recognized with its help. We also prove several easy-to-check necessary conditions for a partition to be a vertex.
Like the Coxeter graph becoming reattached into the Klein graph in [3], the Levi graphs of the \(9_3\) and \(10_3\) self-dual configurations, known as the Pappus and Desargues (\(k\)-transitive) graphs \(\mathcal{P}\) and \(\mathcal{D}\) (where \(k = 3\)), also admit reattachments of the distance-(\(k – 1\)) graphs of half of their oriented shortest cycles via orientation assignments on their common (\(k – 1\))-arcs, concurrent for \(\mathcal{P}\) and opposite for \(\mathcal{D}\), now into 2 disjoint copies of their corresponding Menger graphs. Here, \(\mathcal{P}\) is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while \(\mathcal{D}\) is one of \(7\) CDT graphs with the opposite-reattachment behavior, including the Coxeter graph. Thus, \(\mathcal{P}\) and \(\mathcal{D}\) confront each other in these respects, obtained via \(\mathcal{C}\)-ultrahomogeneous graph techniques \([4,5]\) that allow us to characterize the obtained reattachment Menger graphs in the same terms.
Let \( k \) be a positive integer and \( G = (V, E) \) be a graph of minimum degree at least \( k – 1 \). A function \( f: V \to \{-1, 1\} \) is called a \({signed \; k -dominating\; function}\) of \( G \) if \( \sum_{u \in N_G[v]} f(u) \geq k \) for all \( v \in V \). The \({signed \; k -domination \;number}\) of \( G \) is the minimum value of \( \sum_{v \in V} f(v) \) taken over all signed \( k \)-dominating functions of \( G \). The \({signed \;total \; k-dominating \;function}\) and \({signed\; total \; k -domination\; number}\) of \( G \) can be similarly defined by changing the closed neighborhood \( N_G[v] \) to the open neighborhood \( N_G(v) \) in the definition. The upper \({signed \; k -domination \;number}\) is the maximum value of \( \sum_{u \in V} f(u) \) taken over all \({minimal}\) signed \( k \)-dominating functions of \( G \). In this paper, we study these graph parameters from both algorithmic complexity and graph-theoretic perspectives. We prove that for every fixed \( k \geq 1 \), the problems of computing these three parameters are all \( \mathcal{NP} \)-hard. We also present sharp lower bounds on the signed \( k \)-domination number and signed total \( k \)-domination number for general graphs in terms of their minimum and maximum degrees, generalizing several known results about signed domination.
In this paper, we study a pair of simplicial complexes, which we denote by \( \mathcal{B}(k,d) \) and \( \mathcal{ST}(k+1,d-k-1) \), for all nonnegative integers \( k \) and \( d \) with \( 0 \leq k \leq d-2 \). We conjecture that their underlying topological spaces \( |\mathcal{B}(k,d)| \) and \( |\mathcal{ST}(k+1,d-k-1)| \) are homeomorphic for all such \( k \) and \( d \). We answer this question when \( k = d-2 \) by relating the complexes through a series of well-studied combinatorial operations that transform a combinatorial manifold while preserving its PL-homeomorphism type.
Let \( D = (V,A) \) be a finite and simple digraph. A Roman dominating function (RDF) on \( D \) is a labeling \( f: V(D) \to \{0,1,2\} \) such that every vertex \( v \) with label \( 0 \) has a vertex \( w \) with label \( 2 \) such that \( wv \) is an arc in \( D \). The weight of an RDF \( f \) is the value \( \omega(f) = \sum_{v \in V} f(v) \). The Roman domination number of a digraph \( D \), denoted by \( \gamma_R(D) \), equals the minimum weight of an RDF on \( D \). The Roman reinforcement number \( r_R(D) \) of a digraph \( D \) is the minimum number of arcs that must be added to \( D \) in order to decrease the Roman domination number. In this paper, we initiate the study of Roman reinforcement number in digraphs and we present some sharp bounds for \( r_R(D) \). In particular, we determine the Roman reinforcement number of some classes of digraphs.
Partially filled \(6 \times 6\) Sudoku grids are categorized based on the arrangement of the values in the first three rows. This categorization is then employed to determine the number of \(6 \times 6\) Sudoku grids.
Stankova and West proved in 2002 that the patterns \( 231 \) and \( 312 \) are shape-Wilf-equivalent. Their proof was nonbijective. We give a new characterization of \( 231 \) and \( 312 \) avoiding full rook placements and use this to give a simple bijection that demonstrates the shape-Wilf-equivalence.
The paper begins with a simple circular lock problem that shows how the Combinatorial Nullstellensatz relates to the discrete Fourier Transform.Specifically, the lock shows a relationship between detecting perfect matchings in bipartite graphs using the Combinatorial Nullstellensatz and detecting a maximum rank independent set in the intersection of two matroids in the Fourier transform of a specially chosen function. Finally, an application of the uncertainity principle computes a lower bound for the product of perfect matchings and the number of independent sets.
A \({magic\; square}\) of order \(n\) is an \(n \times n\) array of integers from \(1, 2, \ldots, n^2\) such that the sum of the integers in each row, column, and diagonal is the same number. Two magic squares are \({equivalent}\) if one can be obtained from the other by rotation or reflection. The \({complement}\) of a magic square \(M\) of order \(n\) is obtained by replacing every entry \(a\) with \(n^2 + 1 – a\), yielding another magic square. A magic square is \({self-complementary}\) if it is equivalent to its complement. In this paper, we prove a structural theorem characterizing self-complementary magic squares and present a method for constructing self-complementary magic squares of even order. Combining this construction with the structural theorem and known results on magic squares, we establish the existence of self-complementary magic squares of order \(n\) for every \(n \geq 3\).
Let \(G\) be a graph on \(n\) vertices. If for any ordered set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\), where the vertices in \(S\) appear in the sequence order \(v_1, v_2, \ldots, v_k\), there exists a \(v_1-v_k\) (Hamiltonian) path containing \(S\) in the given order, then \(G\) is \(k\)-ordered (Hamiltonian) connected. In this paper, we show that if \(G\) is \((k+1)\)-connected and \(k\)-ordered connected, then for any ordered set \(S\), there exists a \(v_1-v_k\) path \(P\) containing \(S\) in the given order such that \(|P| \geq \min\{n, \sigma_2(G) – 1\}\), where \(\sigma_2(G) = \min\{d_G(u) + d_G(v) : u,v \in V(G); uv \notin E(G)\}\) when \(G\) is not complete, and \(\sigma_2(G) = \infty\) otherwise. Our result generalizes several related results known before.
Let \(G\) be a simple graph. The incidence energy ( \(IE\) for short ) of \(G\) is defined as the sum of the singular values of the incidence matrix. In this paper, a new lower bound for \(IE\) of graphs in terms of the maximum degree is given. Meanwhile, an upper bound and a lower bound for \(IE\) of the subdivision graph and the total graph of a regular graph \(G\) are obtained, respectively.
The Hosoya polynomial of a graph \(G\) with vertex set \(V(G)\) is defined as \(H(G, z) = \sum_{u,v \in V(G)} x^{d_G(u,v)}\), where \(d_G(u,v)\) is the distance between vertices \(u\) and \(v\). A toroidal polyhex \(H(p,q,t)\) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, described by a string \((p,q,t)\) of three integers \((p \geq 2, q \geq 1, 0 \leq t \leq p-1)\). In this paper, we derive an analytical formula for calculating the Hosoya polynomial of \(H(p,q,t)\) for \(t = 0\) or \(p\leq 2q\) or \(p \leq q+t\). Notably, some earlier results in [2, 6, 26] are direct corollaries of our main findings.
Kotani and Sunada introduced the oriented line graph as a tool in the study of the Ihara zeta function of a finite graph. The spectral properties of the adjacency operator on the oriented line graph can be linked to the Ramanujan condition of the graph. Here, we present a partial characterization of oriented line graphs in terms of forbidden subgraphs. We also give a Whitney-type result, as a special case of a result by Balof and Storm, establishing that if two graphs have the same oriented line graph, they are isomorphic.
Let \(A\) be the \((0,1)\)-adjacency matrix of a simple graph \(G\), and \(D\) be the diagonal matrix \(diag(d_1, d_2, \ldots, d_n)\), where \(d_i\) is the degree of the vertex \(v_i\). The matrix \(Q(G) = D + A\) is called the signless Laplacian of \(G\). In this paper, we characterize the extremal graph for which the least signless Laplacian eigenvalue attains its minimum among all non-bipartite unicyclic graphs with given order and diameter.
In this paper, we investigate some commutativity conditions and extend a remarkable result of Ram Awtar, when Lie ideal \(U\) becomes the part of the centre of \(M\) \(A\)-semiring \(R\).
A pebbling move involves removing two pebbles from one vertex and placing one on an adjacent vertex. The optimal pebbling number of a graph \(G\), denoted by \(f_{opt}(G)\), is the least positive integer \(n\) such that \(n\) pebbles are placed suitably on vertices of \(G\) and, for any specified vertex \(v\) of \(G\), one pebble can be moved to \(v\) through a sequence of pebbling moves. In this paper, we determine the optimal pebbling number of the square of paths and cycles.
In this paper, we verify the list edge coloring conjecture for pseudo- outerplanar graphs with maximum degree at least \(5\) and the equitable \(\Delta\)-coloring conjecture for all pseudo-outerplanar graphs.
We prove that the Cartesian product of two directed cycles of lengths \(n_1\) and \(n_2\) contains an antidirected Hamilton cycle, and hence is decomposable into antidirected Hamilton cycles, if and only if \(\gcd(n_1, n_2) = 2\). For the Cartesian product of \(k > 2\) directed cycles, we establish new sufficient conditions for the existence of an antidirected Hamilton cycle.
Let \(T\) be a tree with no vertices of degree \(2\) and at least one vertex of degree \(3\) or more. A Halin graph \(G\) is a plane graph obtained by connecting the leaves of \(T\) in the cyclic order determined by the planar drawing of \(T\). Let \(\Delta\), \(\lambda(G)\), and \(\chi(G^2)\) denote, respectively, the maximum degree, the \(L(2,1)\)-labeling number, and the chromatic number of the square of \(G\). In this paper, we prove the following results for any Halin graph \(G\): (1) \(\chi(G^2) \leq \Delta + 3\), and moreover \(\chi(G^2) = \Delta + 1\) if \(\Delta \geq 6\); (2) \(\lambda(G) \leq \Delta + 7\), and moreover \(\lambda(G) \leq \Delta + 2\) if \(\Delta \geq 9\).
In this paper, we investigate the zero divisor graph \(G_I(P)\) of a poset \(P\) with respect to a semi-ideal \(I\). We show that the girth of \(G_I(P)\) is \(3\), \(4\), or \(\infty\). In addition, it is shown that the diameter of such a graph is either \(1\), \(2\), or \(3\). Moreover, we investigate the properties of a cut vertex in \(G_I(P)\) and study the relation between semi-ideal \(I\) and the graph \(G_I(P)\), as established in (Theorem 3.9).
A graph \(G\) is \({super-connected}\), or \({super-\(\kappa\)}\), if every minimum vertex-cut isolates a vertex of \(G\). Similarly, \(G\) is \({super-restricted \;edge-connected}\), or \({super-\(\lambda’\)}\), if every minimum restricted edge-cut isolates an edge. We consider the total graph \(T(G)\) of \(G\), which is formed by combining the disjoint union of \(G\) and the line graph \(L(G)\) with the lines of the subdivision graph \(S(G)\); for each line \(l = (u,v)\) in \(G\), there are two lines in \(S(G)\), namely \((l,u)\) and \((l,v)\). In this paper, we prove that \(T(G)\) is super-\(\kappa\) if \(G\) is super-\(\kappa\) graph with \(\delta(G) \geq 4\). \(T(G)\) is super-\(\lambda’\) if \(G\) is \(k\)-regular with \(\kappa(G) \geq 3\). Furthermore, we provide examples demonstrating that these results are best possible.
The paper construct infinite classes of non-isomorphic \(3\)-connected simple graphs with the same total genus polynomial, using overlap matrix, symmetry and Gustin representation. This answers a problem (Problem \(3\) of Page \(38\)) of L.A. McGeoch in his PHD thesis.
The result is helpful for firms to make marketing decisions by calculating the graphs of user demand relationships of different complex ecosystems of platform products and comparing genus polynomials.
A necessary and sufficient condition of the complement to be cordial and its application are obtained.
In this paper, we introduce the notion of blockwise-bursts in array codes equippped with m-metric \([13]\) and obtain some bounds on the parameters of $m$-metric array codes for the detection and correction of blockwise-burst array errors.
Let \(G\) be a graph, and let \(a\) and \(b\) be integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(G)\). In this paper, we obtain a sufficient condition for a graph to have \([a, b]\)-factors including given edges, extending a well-known sufficient condition for the existence of a \(k\)-factor.
We introduce the domination polynomial of a graph \(G\). The domination polynomial of a graph \(G\) of order \(n\) is defined as \(D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma(G)\) is the domination number of \(G\). We obtain some properties of \(D(G, x)\) and its coefficients, and compute this polynomial for specific graphs.
For a tree \(T\), \(Leaf(T)\) denotes the set of leaves of \(T\), and \(T – Leaf(T)\) is called the stem of \(T\). For a graph \(G\) and a positive integer \(m\), \(\sigma_m(G)\) denotes the minimum degree sum of \(m\) independent vertices of \(G\). We prove the following theorem: Let \(G\) be a connected graph and \(k \geq 2\) be an integer. If \(\sigma_3(G) \geq |G| – 2k + 1\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves.
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most \(1\). The equitable chromatic threshold of a graph \(G\), denoted by \(\chi_m^*(G)\), is the minimum \(k\) such that \(G\) is equitably \(k’\)-colorable for all \(k’ > k\). Let \(G \times H\) denote the direct product of graphs \(G\) and \(H\). For \(n \geq m \geq 2\), we prove that \(\chi_m^*(K_m \times K_n)\) equals \(\left\lceil \frac{mn}{m+1} \right\rceil\) if \(n \equiv 2, \ldots, m \pmod{m+1}\), and equals \(m\left\lceil \frac{n}{s^*} \right\rceil\) if \(n \equiv 0, 1 \pmod{m+1}\), where \(s^*\) is the minimum positive integer such that \(s^* \nmid n\) and \(s^* \geq m+2\).
For an undirected graph \(G\) and a natural number \(n\), a \(G\)-design of order \(n\) is an edge partition of the complete graph \(K_n\) with \(n\) vertices into subgraphs \(G_1, G_2, \ldots\), each isomorphic to \(G\). A set \(T \subset V(K_n)\) is called a blocking set if it intersects the vertex set \(V(G_i)\) of each \(G_i\) in the decomposition but contains none of them. Extending previous work [J. Combin. Designs \(4 (1996), 135-142]\), where the authors proved that cycle designs admit no blocking sets, we establish that this result holds for all graphs \(G\). Furthermore, we show that for every graph \(G\) and every integer \(k \geq 2\), there exists a non-\(k\)-colorable \(G\)-design.
Let \(G\) be a planar graph with maximum degree \(\Delta(G)\). The least integer \(k\) such that \(G\) can be partitioned into \(k\) edge-disjoint forests, where each component is a path of length at most \(2\), is called the linear \(2\)-arboricity of \(G\), denoted by \(la_2(G)\). We establish new upper bounds for the linear \(2\)-arboricity of certain planar graphs.
A graph \(G\) of order \(n\) is called a bicyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n+ 1\). In this paper, we study the lexicographic ordering of bicyclic graphs by spectral moments. For each of the three basic types of bicyclic graphs on a fixed number of vertices maximal and minimal graphs in the mentioned order are determined.
An edge irregular total \(k\)-labeling of a graph \(G = (V, E)\) is a labeling \(f: V \cup E \to \{1, 2, \ldots, k\}\) such that the total edge-weights \(wt(xy) = f(x) + f(xy) + f(y)\) are distinct for all pairs of distinct edges. The minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling is called the total edge irregularity strength of \(G\). In this paper, we determine the exact value of the total edge irregularity strength of the Cartesian product of two paths \(P_n\) and \(P_m\). Our result provides further evidence supporting a recent conjecture of Ivančo and Jendrol.
For a vertex set \(S\) with cardinality at least \(2\) in a graph \(G\), a tree connecting \(S\), known as a Steiner tree or \(S\)-tree, is required. Two \(S\)-trees \(T\) and \(T’\) are internally disjoint if \(V(T) \cap V(T’) = S\) and \(E(T) \cap E(T’) = \emptyset\). Let \(\kappa_G(G)\) denote the maximum number of internally disjoint Steiner trees connecting \(S\) in \(G\). The generalized \(k\)-connectivity \(\kappa_k(G)\) of \(G\), introduced by Chartrand et al., is defined as \(\min_{S \subseteq V(G), |S|=k} \kappa_G(S)\). This paper establishes a sharp upper bound for generalized \(k\)-connectivity. Furthermore, graphs of order \(n\) with \(\kappa_3(G) = n-2,n-3\) are characterized.
A hypergraph \(\mathcal{H}\) is said to be \(p\)-Helly when every \(p\)-wise intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has nonempty total intersection. Such hypergraphs were characterized by Berge and Duchet in 1975, and since then they have appeared in various contexts, particularly for \(p=2\), where they are known as Helly hypergraphs. An interesting generalization due to Voloshin considers both the number of intersecting sets and their intersection sizes: a hypergraph \(\mathcal{H}\) is \((p,q,s)\)-Helly if every \(p\)-wise \(q\)-intersecting partial hypergraph \(\mathcal{H}’\) of \(H\) has total intersection of cardinality at least \(s\). This work proposes a characterization for \((p,q,s)\)-Helly hypergraphs, leading to an efficient algorithm for recognizing such hypergraphs when \(p\) and \(q\) are fixed parameters.
A \(k\)-chromatic graph \(G\) is \(uniquely\) \(k\)-\(colorable\) if \(G\) has only one \(k\)-coloring up to permutation of the colors. In this paper, we focus on uniquely \(k\)-colorable graphs on surfaces. Let \({F}^2\) be a closed surface, excluding the sphere, and let \(\chi({F}^2)\) denote the maximum chromatic number of graphs embeddable on \({F}^2\). We shall prove that the number of uniquely \(k\)-colorable graphs on \({F}^2\) is finite if \(k \geq 5\), and characterize uniquely \(\chi({F}^2)\)-colorable graphs on \({F}^2\). Moreover, we completely determine uniquely \(k\)-colorable graphs on the projective plane for \(k \geq 5\).
Given a distribution \(D\) of pebbles on the vertices of a graph \(G\), a pebbling move consists of removing two pebbles from a vertex and placing one on an adjacent vertex (the other is discarded). The pebbling number of a graph, denoted by \(f(G)\), is the minimal integer \(k\) such that any distribution of \(k\) pebbles on \(G\) allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we calculate the pebbling number of the graph \(D_{n,C_m}\) and consider the relationship the pebbling number between the graph \(D_{n,C_m}\) and the subgraphs of \(D_{n,C_m}\).
Let \(G\) and \(H\) be two graphs. A proper vertex coloring of \(G\) is called a dynamic coloring if, for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic coloring with \(k\) colors is denoted by \(\chi_2(G)\). We denote the Cartesian product of \(G\) and \(H\) by \(G \square H\). In this paper, we prove that if \(G\) and \(H\) are two graphs and \(\delta(G) \geq 2\), then \(\chi_2(G \square H) \leq \max(\chi_2(G), \chi(H))\). We show that for every two natural numbers \(m\) and \(n\), \(m, n \geq 2\), \(\chi_2(P_m \square P_n) = 4\). Additionally, among other results, it is shown that if \(3\mid mn\), then \(\chi_2(C_m \square C_n) = 3\), and otherwise \(\chi_2(C_m \square C_n) = 4\).
In \([1]\), Hosam Abdo and Darko Dimitrov introduced the total irregularity of a graph. For a graph \(G\), it is defined as
\[\text{irr}_t(G) =\frac{1}{2} \sum_{{u,v} \in V(G)} |d_G(u) – d_G(v)|,\]
where \(d_G(u)\) denotes the vertex degree of a vertex \(u \in V(G)\). In this paper, we introduce two transformations to study the total irregularity of unicyclic graphs and determine the graph with the maximal total irregularity among all unicyclic graphs with \(n\) vertices.
We consider a variation on the Tennis Ball Problem studied by Mallows-Shapiro and Merlini, \(et \;al\). The solution to the original problem is the well known Catalan numbers, while the variations discussed in this paper yield the Motzkin numbers and other related sequences. For this variation, we present a generating function for the sum of the labels on the balls.
A graph \(G\) of order \(n\) is called a tricyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n + 2\). Let \(\mathcal{T}_n\) denote the set of all tricyclic graphs on \(n\) vertices. In this paper, we determine the first to nineteenth largest Laplacian spectral radii among all graphs in the class \(\mathcal{T}_n\) (for \(n \geq 11\)), together with the corresponding graphs.
The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we determine the lower bounds for the Hosoya index of unicyclic graph with a given diameter. The corresponding extrenal graphs are characterized.
A subset \(S\) of vertices of a graph \(G\) is called a global connected dominating set if \(S\) is both a global dominating set and a connected dominating set. The global connected domination number, denoted by \(\gamma_{gc}(G)\), is the minimum cardinality of a global connected dominating set of \(G\). In this paper, sharp bounds for \(\gamma_{gc}\) are supplied, and all graphs attaining those bounds are characterized. We also characterize all graphs of order \(n\) with \(\gamma_{gc} = k\), where \(3 \leq k \leq n-1\). Exact values of this number for trees and cycles are presented as well.
Let \(\mathbb{F}_q^n\) denote the \(n\)-dimensional row vector space over the finite field \(\mathbb{F}_q\), where \(n \geq 2\). An \(l\)-partial linear map of \(\mathbb{F}_q^n\) is a pair \((V, f)\), where \(V\) is an \(l\)-dimensional subspace of \(\mathbb{F}_q^n\) and \(f: V \to \mathbb{F}_q^n\) is a linear map. Let \(\mathcal{L}\) be the set of all partial linear maps of \(\mathbb{F}_q^n\) containing \(1\). Ordered \(\mathcal{L}\) by ordinary and reverse inclusion, two families of finite posets are obtained. This paper proves that these posets are lattices, discusses their geometricity, and computes their characteristic polynomials.
A total coloring of a graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring \(h\) of a simple graph \(G = (V, E)\) is a proper total coloring of \(G\) such that \(H(u) \neq H(v)\) for any two adjacent vertices \(u\) and \(v\), where \(H(u) = \{h(wu) \mid wu \in E(G)\} \cup \{h(u)\}\) and \(H(v) = \{h(xv) \mid xv \in E(G)\} \cup \{h(v)\}\). The minimum number of colors required for a proper total coloring (resp. an adjacent vertex-distinguishing total coloring) of \(G\) is called the total chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of \(G\) and denoted by \(\chi_t(G)\) (resp. \(\chi_{at}(G)\)). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\), \(\chi(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2\). \(G\) is called Type 1 (resp. Type 2) if \(\chi_t(G) = \Delta(G) + 1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the augmented cube \(AQ_n\) is of Type 1 for \(n \geq 4\). We also consider the adjacent vertex-distinguishing total chromatic number of \(AQ_n\) and prove that \(\chi_{at}(AQ_n) = \Delta(AQ_n) + 2\) for \(n \geq 3 \).
The Channel Assignment Problem is often modeled by integer vertex-labelings of graphs. We will examine \(L(2,1)\)-labelings that realize the span \(\lambda\) of a simple, connected graph \(G = (V, E)\). We define the utility of \(G\) to be the number of possible expansions that can occur on \(G\), where an expansion refers to an opportunity to add a new vertex \(u\) to \(G\), with label \(\lambda(u)\), such that:
Building upon results of Griggs, Jin, and Yeh, we use known values of \(\lambda\) to compute utility for several infinite families and analyze the utility of specific graphs that are of interest elsewhere.
A Sidon set \(S\) is a set of integers where the number of solutions to any integer equation \(k = k_1 + k_2\) with \(k_1, k_2 \in S\) is at most \(2\). If \(g \geq 2\), the set \(S\) is a generalized Sidon set. We consider Sidon sets modulo \(n\), where the solutions to addition of elements are considered under a given modulus. In this note, we give a construction of a generalized Sidon set modulo \(n\) from any known Sidon set.
In an ordered graph \(G\), a set of vertices \(S\) with a pre-coloring of the vertices of \(S\) is said to be a greedy defining set (GDS) if the greedy coloring of \(G\) with fixed colors of \(S\) yields a \(\chi(G)\)-coloring of \(G\). This concept first appeared in [M. Zaker, Greedy defining sets of graphs, Australas. J. Combin, 2001]. The smallest size of any GDS in a graph \(G\) is called the greedy defining number of \(G\). We show that determining the greedy defining number of bipartite graphs is an NP-complete problem, affirmatively answering a problem mentioned in a previous paper. Additionally, we demonstrate that this number for forests can be determined in linear time. Furthermore, we present a method for obtaining greedy defining sets in Latin squares and, using this method, show that any \(n \times n\) Latin square has a GDS of size at most \(n^2 – (n \log 4n)/4\).
Multi-receiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify authenticity of the received message. In this paper, we construct one multi-receiver authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of this codes are also computed.
Resistance distance was introduced by Klein and Randic as a generalization of the classical distance. The Kirchhoff index \(Kf(G)\) of a graph \(G\) is the sum of resistance distances between all pairs of vertices. In this paper, we determine the bicyclic graph of order \(n \geq 8\) with maximal Kirchhoff index. This improves and extends an earlier result by Zhang \(et\; al. [19]\).
Bereg and Wang defined a new class of highly balanced \(d\)-ary trees which they call \(k\)-trees; these trees have the interesting property that the internal path length and thus the Wiener index can be calculated quite easily. A \(k\)-tree is characterized by the property that all levels, except for the last \(k\) levels, are completely filled. Bereg and Wang claim that the number of \(k\)-trees is exponentially increasing, but do not give an asymptotic formula for it. In this paper, we study the number of \(d\)-ary \(k\)-trees and the number of mutually non-isomorphic \(d\)-ary \(k\)-trees, making use of a technique due to Flajolet and Odlyzko.
A group \(G\) is said to be a \(B_k\)-group if for any \(k\)-subset \(\{a_1, \ldots, a_k\}\) of \(G\), \(\left|\{a_ia_j \mid 1 \leq i, j \leq k\}\right| \leq \frac{k(k+1)}{2}\). In this paper, a complete classification of \(B_5\)-groups is given.
In this paper, we give a complete solution to the existence of lattice group divisible \(3\)-designs with block sizes four and six.
Let \( R \) be a commutative ring with identity and \( \mathbb{A}^*(R) \) be the set of non-zero ideals with non-zero annihilators. The annihilating-ideal graph of \( R \) is defined as the graph \( \mathbb{AG}(R) \) with the vertex set \( \mathbb{A}^*(R) \) and two distinct vertices \( I_1 \) and \( I_2 \) are adjacent if and only if \( I_1 I_2 = (0) \). In this paper, we study some connections between the graph-theoretic properties of \( \mathbb{AG}(R) \) and algebraic properties of the commutative ring \( R \).
Let \( A_n = (a_1, a_2, \ldots, a_n) \) and \( B_n = (b_1, b_2, \ldots, b_n) \) be two sequences of nonnegative integers satisfying \( a_1 \geq a_2 \geq \cdots \geq a_n \), \( a_i \leq b_i \) for \( i = 1,2,\ldots,n \), and \( a_i = a_{i+1} \) implies that \( b_i \geq b_{i+1} \) for \( i = 1,2,\ldots,n-1 \). Let \( I \) be a subset of \( \{1,2,\ldots,n\} \) and \( a_i \equiv b_i \pmod{2} \) for each \( i \in I \). \( (A_n; B_n) \) is said to be partial parity graphic with respect to \( I \) if there exists a simple graph \( G \) with vertices \( v_1, v_2, \ldots, v_n \), such that \( a_i \leq d_G(v_i) \leq b_i \) for \( i = 1,2,\ldots,n \) and \( d_G(v_i) \equiv b_i \pmod{2} \) for each \( i \in I \). In this paper, we give a characterization for \( (A_n; B_n) \) to be partial parity graphic. This is a variation of the partial parity \( (g, f) \)-factor theorem due to Kano and Matsuda in degree sequences.
It’s well known that all of the pooling designs constructed are based on a finite set or a finite vector space. In this paper, we construct two families of pooling designs not only based on finite sets (resp. finite vector spaces) but also on partial mappings (resp. partial linear mappings), and discuss their error-tolerance properties.
Let \( 2^{[m]} \) be ordered by set inclusion, and let \( \mathcal{B} \subseteq 2^{[m]} \) be an antichain. An antichain \( \mathcal{B} \) is called \( k \)-regular (\( k \in \mathbb{N} \)) if for each \( i \in [m] \) there are exactly \( k \) blocks \( B_1, B_2, \ldots, B_k \in \mathcal{B} \) containing \( i \). An antichain is called flat if there exists a positive integer \( l \) such that \( l \leq |B| \leq l+1 \) for all \( B \in \mathcal{B} \), and we call an antichain maximal if the collection of sets \( \mathcal{B} \cup \{B\} \) is not an antichain for all \( B \notin \mathcal{B} \). We call a maximal \( k \)-regular antichain \( \mathcal{B} \subseteq \binom{[m]}{2} \cup \binom{[m]}{3} \) a \( (k,m) \)-MFRAC. In this paper we analyze \( (k,m) \)-MFRACs in the cases \( m \leq 7 \), \( k = m \), \( k = m-1 \), and \( k = m-2 \). We provide some constructions, give necessary conditions for existence, and mention some open problems.
Generalized binomial coefficients are considered. The aim of this paper is to provide a new general combinatorial interpretation of the Lucas-nomial and \( (p,q) \)-nomial coefficients in terms of tiling of \( d \)-dimensional rectangular boxes. The recurrence relation of these numbers is proved in a combinatorial way. To this end, our results are extended to the case of corresponding multi-nomial coefficients.
In this paper, we determine the necessary and sufficient conditions for the existence of simple incomplete triple systems for all \( \lambda \leq 6 \).
Dudeney’s round table problem asks for a set of Hamilton cycles in \( K_n \), having the property that each \( 2 \)-path in \( K_n \) lies in exactly one of the cycles. In this paper, we show how to construct a solution of Dudeney’s round table problem for even \( n \) from a semi-antipodal Hamilton decomposition of \( K_{n-1} \).
Using the spectral invariants of graphs, we present sufficient conditions for some stable properties of graphs.
A matching \( M \) in a graph \( G \) is a subset of \( E(G) \) in which no two edges have a vertex in common. A vertex \( V \) is unsaturated by \( M \) if there is no edge of \( M \) incident with \( V \). A matching \( M \) is called a perfect matching if there is no vertex of the graph that is unsaturated by \( M \). Let \( G \) be a \( k \)-edge-connected graph, \( k \geq 1 \), on even \( n \) vertices, with minimum degree \( r \) and maximum degree \( r + e \), \( e \geq 1 \). In this paper, we find a lower bound for \( n \) when \( G \) has no perfect matchings.
Two kinds of authentication schemes are constructed using singular symplectic geometry over finite fields in this paper. One is an authentication code with arbitration, another is a multi-receiver authentication code. The parameters of two kinds of codes have been computed. Under the assumption that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the maximum probabilities of success of different types of deception attacks are also computed.
If \( G_1 \) and \( G_2 \) are two graphs, then the edge amalgamation \( G_1 *_e G_2 \) is defined to be the graph obtained by identifying some given edge of \( G_1 \) with some given edge of \( G_2 \). In this paper, it is shown that \( \gamma(G_1 *_e G_2 *_e \ldots *_e G_n) = \lceil \frac{n}{2} \rceil \) where \( G_i \) (\( 1 \leq i \leq n \)) is a critical graph of minimum genus \( 1 \).
A set \( S \subseteq V \) is a dominating set of a graph \( G = (V, E) \) if each vertex in \( V \) is either in \( S \) or is adjacent to a vertex in \( S \). A vertex is said to dominate itself and all its neighbors. A set \( S \subseteq V \) is a \({total \;dominating\; set}\) of a graph \( G = (V, E) \) if each vertex in \( V \) is adjacent to a vertex in \( S \). In total domination, a vertex no longer dominates itself. These two types of domination can be thought of as representing the vertex set of a graph as the union of the closed (domination) and open (total domination) neighborhoods of the vertices in the set \( S \). A set \( S \subseteq V \) is a \({total, efficient\; dominating\; set}\) (also known as an \({efficient \;open\; dominating \;set}\)) of a graph \( G = (V, E) \) if each vertex in \( V \) is adjacent to exactly one vertex in \( S \). In 2002, Gavlas and Schultz completely classified all cycle graphs that admit a total, efficient dominating set. This paper extends their result to two classes of Cayley graphs.
Seymour’s Second Neighborhood Conjecture claims that every simple digraph has a vertex whose first neighborhood is at most as large as its second neighborhood. We confirm this conjecture for neighbor-connection free simple digraphs and distance-two simple digraphs. As a consequence, the conjecture is true for triangle-free digraphs and \(4\)-cycle free digraphs.
We consider the random process arising from a sequence of random Seidel switching operations on \( n \) vertices. We show that this process can be interpreted as a random walk on a Cayley graph of an abelian group, and use spectral methods to show that the random process converges to a stationary distribution in \( O(n \log(n)) \) steps. We then consider two generalizations: we allow multiple states for each edge, and restrict the process to a fixed host graph \( H \). We then analyze the general case and obtain convergence results for any graph \( H \).
In this paper, we present the study of the interlace polynomials for \( n \)-claw graphs. For a positive integer \( n > 1 \), an \( n \)-claw graph \( W_n \) is a tree that has one center vertex and \( n \) claws. The center vertex is connected to one vertex of each of the \( n \) claws using one edge of the claw. We present iterative formulas and explicit formulas for the interlace polynomial of \( W_n \). Furthermore, some interesting properties of the polynomial are discussed.
Bar visibility graphs (BVG) are graphs whose vertices can be assigned disjoint horizontal line segments in the plane so that adjacent vertices correspond to pairs of bars that are visible to each other via an unobstructed, vertical band of visibility. A \( k \)-stack layout of a graph is a linear vertex ordering and a \( k \)-edge coloring such that each color class avoids crossing edges with respect to the linear order. BVGs and stack layouts were introduced separately in the 1970s and have many applications including testing circuit boards, VLSI design, and graph drawing. Motivated by applications to carousel navigation design, we introduce a hybrid class of graphs called unit stack visibility graphs and give a combinatorial characterization of these graphs. We leave open the problem of determining whether a polynomial-time algorithm exists to recognize unit stack visibility graphs.
Two quasigroup identities of importance in combinatorics, Schröder’s Second Law and Stein’s Third Law, share many common features that are incorporated under the guise of palindromic quasigroups. A graph-theoretical technique yields a topological proof for the congruence restrictions on the spectrum of Schröder or outer palindromic quasigroups. The potential for a comparable proof applicable to Stein or inner palindromic quasigroups raises open graph-theoretical and combinatorial problems. Imposition of extra Sudoku-like conditions on Latin squares of square order, based on the coloring of so-called Sudoku graphs, leads to the concept of a Sudoku quasigroup. It is shown that the spectrum of inner palindromic Sudoku quasigroups comprises every perfect square, thereby identifying the chromatic number of each Sudoku graph.
Hoffman proved that for a simple graph \( G \), the chromatic number \( \chi(G) \) obeys \( \chi(G) \geq 1 – \frac{\lambda_1}{\lambda_n} \), where \( \lambda_1 \) and \( \lambda_n \) are the maximal and minimal eigenvalues of the adjacency matrix of \( G \), respectively. Lovász later showed that \( \chi(G) \geq 1 – \frac{\lambda_1}{\lambda_n} \) for any (perhaps negatively) weighted adjacency matrix.
In this paper, we give a probabilistic proof of Lovász’s theorem, then extend the technique to derive generalizations of Hoffman’s theorem when allowed a certain proportion of edge-conflicts. Using this result, we show that if a \( 3 \)-uniform hypergraph is \( 2 \)-colorable, then \( \bar{d} \leq – \frac{3}{2} \lambda_{\text{min}} \) where \( \bar{d} \) is the average degree and \( \lambda_{\text{min}} \) is the minimal eigenvalue of the underlying graph. We generalize this further for \( k \)-uniform hypergraphs, for the cases \( k = 4 \) and \( 5 \), by considering several variants of the underlying graph.
In 1967, Erdős and Hajnal asked the question: Does there exist a \( K_4 \)-free graph that is not the union of two triangle-free graphs? Finding such a graph involves solving a special case of the classical Ramsey arrowing operation. Folkman proved the existence of these graphs in 1970, and they are now called Folkman graphs. Erdős offered \$100 for deciding if one exists with less than \( 10^{10} \) vertices. This problem remained open until 1988 when Spencer, in a seminal paper using probabilistic techniques, proved the existence of a Folkman graph of order \( 3 \times 10^9 \) (after an erratum), without explicitly constructing it. In 2008, Dudek and Rödl developed a strategy to construct new Folkman graphs by approximating the maximum cut of a related graph, and used it to improve the upper bound to 941. We improve this bound first to 860 using their approximation technique and then further to 786 with the MAX-CUT semidefinite programming relaxation as used in the Goemans-Williamson algorithm.
Let a set \([n] = \{1,2,\ldots,n\}\) be given. Finding a subset \( S \) of \( 2^{[n]} \) with minimum cardinality such that, for any two distinct elements \( x, y \in [n] \), there exist disjoint subsets \( A_x, A_y \in \mathcal{S} \) such that \( x \in A_x \) and \( y \in A_y \) is called the \emph{extremal set} problem. In this paper, we define the Extremal Set Decision (ESD) Problem and study its complexity.
A cyclic base ordering of a connected graph \( G \) is a cyclic ordering of \( E(G) \) such that every \( |V(G)| – 1 \) cyclically consecutive edges form a spanning tree of \( G \). Let \( G \) be a graph with \( E(G) \neq \emptyset \) and let \( \omega(G) \) denote the number of components in \( G \). The invariants \( d(G) \) and \( \gamma(G) \) are respectively defined as \( d(G) = \frac{|E(G)|}{|V(G)| – \omega(G)} \) and \( \gamma(G) = \text{max}\{d(H)\} \), where \( H \) runs over all subgraphs of \( G \) with \( E(H) \neq \emptyset \). A graph \( G \) is uniformly dense if \( d(G) = \gamma(G) \). Kajitani et al. [8] conjectured in 1988 that a connected graph \( G \) has a cyclic base ordering if and only if \( G \) is uniformly dense. In this paper, we show that this conjecture holds for some classes of uniformly dense graphs.
A graph \( G \) is a \((t, r)\)-regular graph if every collection of \( t \) independent vertices is collectively adjacent to exactly \( r \) vertices. Let \( p, s \), and \( m \) be positive integers, where \( m \geq 2 \), and let \( G \) be a \((2, r)\)-regular graph. If \( n \) is sufficiently large, then \( G \) is isomorphic to \( K_s + mK_p \), where \( 2(p-1) + s = r \). A nested \((2, r)\)-regular graph is constructed by replacing selected cliques in a \((2, r)\)-regular graph with a \((2, r’)\)-regular graph and joining the vertices of the peripheral cliques. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.
Given a graph \( G \), we show how to compute the number of (perfect) matchings in the graphs \( G \Box P_n \) and \( G \Box C_n \), by looking at appropriate entries in a power of a particular matrix. We give some generalizations and extensions of this result, including showing how to compute tilings of \( k \times n \) boards using monomers, dimers, and \( 2 \times 2 \) tiles.
Consider a simple undirected graph \( G = (V, E) \). A family of subtrees, \(\{T_v\}_{v \in V}\), of a tree \(\mathcal{T}\) is called a \((\mathcal{T}; t)\)-representation of \(G\) provided \( uv \in E \) if and only if \( |T_u \cap T_v| \geq t \). In this paper, we consider \((\mathcal{T}; t)\)-representations for graphs containing large asteroidal sets, where \(\mathcal{T}\) is a subdivision of the \(n\)-star \(K_{1, n}\). An asteroidal set in a graph \(G\) is a subset \(A\) of the vertex set such that for all 3-element subsets of \(A\), there exists a path in \(G\) between any two of these vertices which avoids the neighborhood of the third vertex. We construct a representation of an asteroidal set of size \( n + \sum_{k=2}^{n} \binom{n}{k} \binom{t-2}{k-1} \) and show that no graph containing a larger asteroidal set can be represented.
For a simple undirected graph \(G = (V, E)\), a subset \(I\) of \(V(G)\) is said to be an independent set of \(G\) if any two vertices in \(I\) are not adjacent in \(G\). A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we survey the largest to fourth largest numbers of maximal independent sets among all trees and forests. In addition, we further look into the problem of determining the fifth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.
Ruskey and Savage posed the question: For \(n \geq 2\), does every matching in \(Q_n\) extend to a Hamiltonian cycle in \(Q_n\)? Fink showed that the answer is yes for every perfect matching, thereby proving Kreweras’ conjecture. In this paper, we prove that for \(n \geq 3\), every matching in \(Q_n\) not covering exactly two vertices at distance \(3\) extends to a Hamiltonian cycle in \(Q_n\). An edge in \(Q_n\) is an \(i\)-edge if its endpoints differ in the \(i\)th position. We also show that for \(n \geq 2\), every matching in \(Q_n\) consisting of edges in at most four types extends to a Hamiltonian cycle in \(Q_n\).
In this paper, the congruence relations and the lower and upper bounds of hyper-Wiener index for \(k\)-membered ring spiro systems given length \(n\) are determined respectively. As these results’ applications,the congruence relations and the extremal five- and six-membered ring spiro systems with maximal and minimal hyper-Wiener index are given respectively.
Let \(G\) be a finite group and \(S \subseteq G \setminus \{0\}\). We call \(S\) an additive basis of \(G\) if every element of \(G\) can be expressed as a sum over a nonempty subset in some order. Let \(cr(G)\) be the smallest integer \(t\) such that every subset of \(G \setminus \{0\}\) of cardinality \(t\) is an additive basis of \(G\). In this paper, we determine \(cr(G)\) for the following cases: (i) \(G\) is a finite nilpotent group; (ii) \(G\) is a group of even order which possesses a subgroup of index \(2\).
For \(n \geq 1\), we let \(a_n\) count the number of compositions of the positive integer \(m\), where the last summand is odd. We find that \(a_n = (\frac{1}{3})(-1)^n + (\frac{2}{3}) 2^{n-1}\). Since \(J_n\), the \(n\)-th Jacobsthal number, is given as \(\frac{1}{3}(-1)^n + \frac{2}{3}2^{n-1}\) for \(n \geq 0\), it follows that \(a_n = J_{n-1}\) for \(n \geq 1\). For this reason, these compositions are often referred to as the Jacobsthal compositions.
In our investigation, we determine results for the \(a_n\) compositions of \(n\), such as: (i) \(a_{n,k}\), the number of times the positive integer \(k\) appears as a summand among these \(a_n\) compositions of \(n\); (ii) the numbers of plus signs, summands, even summands, and odd summands that occur for these compositions of \(n\); (iii) the sum of the even summands and the sum of the odd summands for the \(a_n\) compositions of \(n\); (iv) the numbers of levels, rises, and descents for the \(a_n\) compositions; and (v) the number of runs that occur among these \(a_n\) compositions.
In this paper, we introduce a new sequence called standard Young words, which are defined as quaternary words with interesting restrictions. First, we show that the cardinality of standard Young words of length n is related to Catalan triangle sequence and we establish a bijection from the set of standard Young words to the set of pairs of non-intersection lattice paths. Then we set a one-to-one correspondence between the set of standard Young words and the set of standard Young tableaux of two rows, which results in the correspondence between the statistics of standard Young words and standard Young tableaux, such as sign and descents.
A graph \(G\) is called a fractional \((k, m)\)-deleted graph if after deleting any \(m\) edges of \(G\), the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for \(k \geq 2\) and \(m \geq 0\), \(G\) is a fractional \((k, m)\)-deleted graph if one of the following conditions holds: 1) \(n \geq 4k + 4m – 3\), \(\delta(G) \geq k + m\), and \(\max\{d_G(u), d_G(v)\} \geq \frac{n}{2}\) for each pair of non-adjacent vertices \(u\) and \(v\) of \(G\); 2) \(\delta(G) \geq k + m\), \(\omega_2(G) \geq n\), \(n \geq 4k + 4m – 5\) if \((k, m) = (3, 0)\), and \(n \geq 8\) if \((k, m) = (3, 0)\). The results are best possible in some sense.
Let \(K\) be a real quadratic field \(\mathbb{Q}(\sqrt{n})\) with an integer \(n = df^2\), where \(d\) is the field discriminant of \(K\) and \(f \geq 1\). Q. Mushtaq found an interesting phenomenon that any totally negative number \(\kappa_0\) with \(\kappa^{\sigma} < 0\) and \(\kappa_0^{\sigma} < 0\) belonging to the discriminant \(n\), attains an ambiguous number \(\kappa_m\) with \(\kappa_m \kappa_m^{\sigma} < 0\) after finitely many actions \(\kappa_0^{A_j}\) with \(0 \leqq j \leqq m\) by modular transformations \(A_j \in \mathrm{SL}_2^+(\mathbb{Z})\). Here \(\sigma\) denotes the embedding of \(K\) distinct from the identity. In this paper, we give a new aspect for the process to reach an ambiguous number from a totally negative or totally positive number, by which the gap of the proof of Q. Mushtaq's Theorem is complemented. Next, as an analogue of Gauss' Genus Theory, we prove that the ring class number \(h_{+}(df^2)\) coincides with the ambiguous class number belonging to the discriminant \(n = df^2\), and its behavior is unbounded when \(f\) with suitable prime factors goes to infinity using the ring class number formula.
For a rational number \(r > 1\), a set \(A\) of positive integers is called an \(r\)-multiple-free set if \(A\) does not contain any solution of the equation \(rx = y\). The extremal problem of estimating the maximum possible size of \(r\)-multiple-free sets contained in \([n] := \{1, 2, \ldots, n\}\) has been studied in combinatorial number theory for theoretical interest and its application to coding theory. Let \(a\) and \(b\) be relatively prime positive integers such that \(a < b\). Wakeham and Wood showed that the maximum size of \((b/a)\)-multiple-free sets contained in \([n]\) is \( \frac{b}{b+1} + O(\log n)\). In this note, we generalize this result as follows. For a real number \(p \in (0, 1)\), let \([n]_p\) be a set of integers obtained by choosing each element \(i \in [n]\) randomly and independently with probability \(p\). We show that the maximum possible size of \((b/a)\)-multiple-free sets contained in \([n]_p\) is \({\frac{b}{b+p}pn} + O(\sqrt{pn} \log n \log \log n)\) with probability that goes to \(1\) as \(n \to \infty\).
A partition of an integer \(n\) is a representation \(n = a_1 + a_2 + \cdots + a_k\), with integer parts \(a_1 \geq a_2 \geq \cdots \geq a_k \geq 1\). The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of \(n\). As a consequence, interesting partition identities are derived. The more general case of Durfee rectangles is also treated, as well as the asymptotic growth of the mean area over all partitions of \(n\).
A graph \(G\) is called a fractional \((k, m)\)-deleted graph if any \(m\) edges are removed from \(G\), then the resulting graph admits a fractional \(k\)-factor. In this paper, we prove that for integers \(k \geq 2\), \(m \geq 0\), \(n \geq 8k + 4m – 7\), and \(\delta(G) \geq k + m\), if
\[|N_G(x) \cup N_G(y)| \geq \frac{n}{2}\]
for each pair of non-adjacent vertices \(x, y\) of \(G\), then \(G\) is a fractional \((k, m)\)-deleted graph. The bounds for neighborhood union condition, order, and the minimum degree of \(G\) are all sharp.
A \(c\)-partite or multipartite tournament is an orientation of a complete \(c\)-partite graph. A digraph \(D\) is cycle complementary if there exist two vertex-disjoint directed cycles \(C\) and \(C’\) such that \(V(D) = V(C) \cup V(C’)\). The global irregularity of a digraph \(D\) is defined by
\[i_g(D) = \max\{\max(d^+(x), d^-(x)) – \min(d^+(y),d^-(y)) \mid x,y \in V(D)\}.\]
If \(i_g(D) = 0\), then \(D\) is regular, and if \(i_g(D) \leq 1\), then \(D\) is almost regular. We prove in this paper that every almost regular \(c\)-partite tournament with \(c \geq 3\) such that all partite sets have the same cardinality \(r \geq 4\) contains two complementary directed cycles of length \(3\) and \(|V(D)| – 3\).
In this paper, we determine the spectrum for \(super-perfect\) OQSs. OQSs are \(G\)-designs in which \(G\) is an octagon quadrangle, i.e., the graph consisting of an \(8\)-cycle \((x_1, x_2, \ldots, x_8)\) with two additional chords: the edges \(\{x_1, x_4\}\) and \(\{x_5, x_6\}\).
In this paper, we give a four parameter theta function identity and prove it by using some properties of Jacobi’s theta functions and Jacobi’s fundamental formulae.
The order dimension is an invariant on partially ordered sets introduced by Dushnik and Miller in \(1941 [1]\). It is known that the computation of the order dimension of a partially ordered set in general is highly complex,with current algorithms relying on the minimal coloring of an associated hypergraph, see \([5]\). The aim of this work is to extend the family of posets whose order dimension is easily determined by a formula. We introduce an operation called layering. Finally, we provide the precise formulas for determining the order dimension of any given number of layers of Trotter’s generalized crowns.
In this paper, the regular endomorphisms of a split graph are investigated. We give a condition under which the regular endomorphisms of a split graph form a monoid.
The clique graph \(K(G)\) of a graph \(G\) is the intersection graph of all its (maximal) cliques, and \(G\) is said to be clique divergent if the order of its \(n\)-th iterated clique graph \(K^n(G)\) tends to infinity with \(n\). In general, deciding whether a graph is clique divergent is not known to be computable. We characterize the dynamical behavior under the clique operator of circulant graphs of the form \(C_n(a, b, c)\) with \(0 < a < b < c < \frac{n}{3}\). Such a circulant is clique divergent if and only if it is not clique-Helly. Owing to the Dragan-Szwarcfiter Criterion to decide clique-Hellyness, our result implies that the clique divergence of these circulants can be decided in polynomial time. Our main difficulty was the case \(C_n(1, 2, 4)\), which is clique divergent but no previously known technique could be used to prove it.
A total dominating set \(S\) of a graph \(G\) with no isolated vertex is a locating-total dominating set of \(G\) if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\) are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian product of cycles \(C_m\) and \(C_n\), and prove that for any positive integer \(n \geq 3\), the locating-total domination numbers of the Cartesian product of cycles \(C_3\) and \(C_n\) is equal to \(n\) for \(n \equiv 0 \pmod{6}\) or \(n + 1\) otherwise.
A graph \(G\) is called a fractional \((g, f, m)\)-deleted graph if after deleting any \(m\) edges, then the resulting graph admits a fractional \((g, f)\)-factor. In this paper, we prove that if \(G\) is a graph of order \(n\), and if \(1 \leq g(x) \leq f(x) \leq 6\) for any \(x \in V(G)\), \(\delta(G) \geq \frac{b^2(i-1)}{a} ++2m\), \(n > \frac{(a+b)(i(a+b)+2m-2)}{a}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_i)| \geq \frac{bn}{a+b} \), for any independent set \(\{x_1, x_2, \ldots,x_i\}\) of \(V(G)\), where \(i \geq 2\), then \(G\) is a fractional \((g, f, m)\)-deleted graph. The result is tight on the neighborhood union condition.
In this short paper, we introduce the second order linear recurrence relation of the \(AB\)-generalized Fibonacci sequence and give the explicit formulas for the sums of the positively and negatively subscripted terms of the \(AB\)-generalized Fibonacci sequence by matrix methods. This sum generalizes the one obtained earlier by Kilig in \([2]\).
Only few results concerning crossing numbers of join of some graphs are known. In the paper, for the special graph \(G\) on six vertices, we give the crossing numbers of \(G\vee P_n\) and \(G\vee C_n\), \(P_n\) and \(C_n\) are the path and cycle on \(n\) vertices, respectively.
Recently, Dere and Simsek have treated some applications of umbral algebra. related to several special polynomials(see \([8]\)). In this paper, we derive some new and interesting identities of special polynomials involving Bernoulli, Euler and Laguerre polynomials arising from umbral calculus.
In this paper, we prove that for any tree \(T\), \(T^2\) is a divisor graph if and only if \(T\) is a caterpillar and the diameter of \(T\) is less than six. For any caterpillar \(T\) and a positive integer \(k \geq 1\) with \(diam(T) \leq 2k\), we show that \(T^k\) is a divisor graph. Moreover, for a caterpillar \(T\) and \(k \geq 3\) with \(diam(T) = 2k\) or \(diam(T) = 2k + 1\), we show that \(T^k\) is a divisor graph if and only if the centers of \(T\) have degree two.
To construct a large graph from two smaller ones that have same order, one can add an arbitrary perfect matching between their vertex-sets. The topologies of many networks are special cases of these graphs. An interesting and important problem is how to persist or even improve their link reliability and link fault-tolerance. Traditionally, this may be done by optimizing the edge connectivity of their topologies, a more accurate method is to improve their \(m\)-restricted edge connectivity. This work presents schemes for optimizing \(m\)- restricted edge connectivity of these graphs, some well-known results are direct consequences of our observations.
In this paper we introduce a new kind of generalized Pell numbers. This generalization is introduced in the distance sense. We give different interpretations and representations of these numbers.We present relations between distance Pell numbers and Fibonacci numbers. Moreover we describe graph interpretations of distance Pell numbers. These graphs interpretations in the natural way imply a new kind of generalized Jacobsthal numbers.
A graph \(G\) is called a fractional \((g, f, n’, m)\)-critical deleted graph if after deleting any \(n’\) vertices of \(G\) the remaining graph is a fractional \((g, f, m)\)-deleted graph. In this paper, we give two binding number conditions for a graph to be a fractional \((g, f, n’, m)\)-critical deleted graph.
In this paper, we compute the hyper-Wiener index of arbitrary \(k\)-membered ring spiro chain. We also determine the extremal \(k\)-membered ring spiro chains for hyper-Wiener index.
In this paper, the notion of cyclic bursts in array codes equipped with a non-Hamming metric \([13]\) as a generalization of classical cyclic bursts \([5]\) is introduced and some bounds are obtained on the parameters of array codes for the detection and correction of cyclic burst array errors.
Let \(G\) be a graph, and let \(a\), \(b\), \(k\) be integers with \(0 \leq a \leq b\), \(k \geq 0\). An \([a, b]\)-factor of graph \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(F)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\) the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, it is proved that, if \(a\), \(b\), \(k\) be integers with \(1 \leq a < b\), \(k \geq 0\) and \(b \geq a(k+1)\) and \(G\) is a graph with \(\delta(G) \geq a+k\) and binding number \(b(G) \geq a-1+\frac{a(k+1)}{b}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.
Let \(R(a(x-y) = bz)\) denote the least integer \(n\) such that for every \(2\)-coloring of the set \(\{1, 2, \ldots, n\}\) there exists a monochromatic solution to \(a(x-y) = bz\). Recently, Gasarch, Moriarty, and Tumma conjectured that \(R(a(x-y) = bz) = b^2 + b + 1\), where \(1 < a < b\). In this note, we confirm this conjecture.
In this paper, we introduce the notion of a generalized triple derivation \(f\), with an associated triple derivation \(d\), on a lattice and investigate some related results. Among some other results, we prove that: Let \((L, \wedge, \vee)\) be a distributive lattice and \(f\) be a generalized triple derivation, with associated triple derivation \(d\), on \(L\). Then the following conditions are equivalent for all \(x, y, z \in L\):
The scrambling index of an \(n \times n\) primitive matrix \(A\) is the smallest positive integer \(k\) such that \(A^k(A^T)^k > 0\), where \(A^T\) denotes the transpose of \(A\). In 2009, M. Akelbek and S. Kirkland gave an upper bound on the scrambling index of an \(n \times n\) primitive matrix \(M\) in terms of its order \(n\), and they also characterized the primitive matrices that achieve the upper bound. In this paper, we characterize primitive matrices which achieve the second largest scrambling index in terms of its order. Meanwhile, we show that there exists a gap in the scrambling index set of primitive matrices.
Let \(d_G(v)\) be the degree of a vertex \(v\) in a graph \(G\). A graph \(G\) is called a \(D(i_1, \ldots,i_k)\) graph, if \(\{d_G(v) \mid x \in V(G)\} = \{i_1, \ldots, i_k\}\). In this paper, a necessary and sufficient condition for a connected \(D(1, 3)\) graph to be cordial is given.
Let \(G\) be a connected graph of order \(n\), and suppose that \(n = \sum_{i=1}^{k}n_i\), where \(n_1, n_2, \ldots,n_n\) are integers with at least two. A spanning subgraph is called a path-factor if each component of it is a path of order at least two. In [Y. Chen, F. Tian, B, Wei, Degree sums and path-factors in graphs, Graphs and Combin. \(17 (2001),61-71.]\), Chen et al. gave a degree sum condition for the existence of a path-factor consisting of paths of order \(n_1, n_2, \ldots, n_k\). In this paper, for 2-connected graphs, we generalize this result.
Let \(G\) be a graph with \(n\) vertices and \(\mu_1, \mu_2, \ldots, \mu_n\) be the Laplacian eigenvalues of \(G\). The Laplacian-energy-like graph invariant \(\text{LEL}(G) = \sum_{i=1}^{n} \sqrt{\mu_i}\) has been defined and investigated in [1]. Two non-isomorphic graphs \(G_1\) and \(G_2\) of the same order are said to be \(\text{LEL}\)-equienergetic if \(\text{LEL}(G_1) = \text{LEL}(G_2)\). In [2], three pairs of \(\text{LEL}\)-equienergetic non-cospectral connected graphs are given. It is also claimed that the \(\text{LEL}\)-equienergetic non-cospectral connected graphs are relatively rare. It is natural to consider the question: Whether the number of the \(\text{LEL}\)-equienergetic non-cospectral connected graphs is finite? The answer is negative, because we shall construct a pair of \(\text{LEL}\)-equienergetic non-cospectral connected graphs of order \(n\), for all \(n \geq 12\) in this paper.
The status of a vertex \(v\) in a graph is the sum of the distances between \(v\) and all vertices. The status sequence of a graph is the list of the statuses of all vertices arranged in nondecreasing order. It is well known that non-isomorphic graphs may have the same status sequence. This paper gives a sufficient condition for a graph \(G\) with the property that there exists another graph \(G’\) such that \(G’\) and \(G\) have the same status sequence and \(G’\) is not isomorphic to \(G\).
We give combinatorial proofs of some binomial and $q$-binomial identities in the literature, such as
\[\sum\limits_{k={-\infty}}^{\infty}(-1)^kq^{\frac{(9k^2+3k)}{2}}\binom{2n}{n+3k}=(1+q^n)\prod\limits_{k=1}^{n-1}(1+q^k+q^{2k})(n\geq 1)\]
and
\[\sum\limits_{k=0}^{\infty} \binom{3n}{2k}(-3)^k=(-8)^n.\]
Two related conjectures are proposed at the end of this paper.
In the spirit of Ryser’s theorem, we prove sufficient conditions on \(k\), \(\ell\), and \(m\) so that \(k \times \ell \times m\) Latin boxes, i.e., partial Latin cubes whose filled cells form a \(k \times \ell \times m\) rectangular box, can be extended to a \(k \times n \times m\) Latin box, and also to a \(k \times n \times m\) Latin box, where \(n\) is the number of symbols used, and likewise the order of the Latin cube. We also prove a partial Evans-type result for Latin cubes, namely that any partial Latin cube of order \(n\) with at most \(n-1\) filled cells is completable, given certain conditions on the spatial distribution of the filled cells.
A star-factor of a graph \(G\) is a spanning subgraph of \(G\) such that each component is a star. An edge-weighting of \(G\) is a function \(w: E(G) \rightarrow \mathbb{N}^+\), where \(\mathbb{N}^+\) is the set of positive integers. Let \(\Omega\) be the family of all graphs \(G\) such that every star-factor of \(G\) has the same weight under some fixed edge-weighting \(w\). The open problem of characterizing the class \(\Omega\), posed by Hartnell and Rall, is motivated by the minimum cost spanning tree and the optimal assignment problems. In this paper, we present a simple structural characterization of the graphs in \(\Omega\) that have girth at least five.
We show that whenever the length four words over a three letter alphabet are two-colored, there must exist a monochromatic combinatorial line. We also provide some computer generated lower bounds for some other Hales-Jewett numbers.
This paper introduces a method for finding closed forms for certain sums involving squares of binomial coefficients. We use this method to present an alternative approach to a problem of evaluating a different type of sums containing squares of the numbers from
Catalan’s triangle.
In this paper, we deal with a special kind of hypergraph decomposition. We show that there exists a decomposition of the 3-uniform hypergraph \(\lambda K_v^{(3)}\) into a special kind of hypergraph \(K_{4}^{(3)} – e\) whose leave has at most two edges, for any positive integers \(v \geq 4 \) and \(\lambda\).
For a connected graph \(G\) of order \(n \geq 2\) and a linear ordering \(s = v_1, v_2, \ldots, v_n\) of \(V(G)\), define \(d(s) = \sum_{i=1}^{n-1} d(v_i, v_{i+1})\), where \(d(v_i, v_{i+1})\) is the distance between \(v_i\) and \(v_{i+1}\). The traceable number \(t(G)\) and upper traceable number \(t^+(G)\) of \(G\) are defined by \(t(G) = \min\{d(s)\}\) and \(t^+(G) = \max\{d(s)\}\), respectively, where the minimum and maximum are taken over all linear orderings \(s\) of \(V(G)\). The traceable number \(t(v)\) of a vertex \(v\) in \(G\) is defined by \(t(v) = \min\{d(s)\}\), where the minimum is taken over all linear orderings \(s\) of \(V(G)\) whose first term is \(v\). The \({maximum\; traceable \;number}\) \(t^*(G)\) of \(G\) is then defined by \(t^*(G) = \max\{t(v) : v \in V(G)\}\). Therefore, \(t(G) \leq t^*(G) \leq t^+(G)\) for every nontrivial connected graph \(G\). We show that \(t^*(G) \leq \lfloor \frac{t(G)+t^+(G)+1}{2}\rfloor\) for every nontrivial connected graph \(G\) and that this bound is sharp. Furthermore, it is shown that for positive integers \(a\) and \(b\), there exists a nontrivial connected graph \(G\) with \(t(G) = a\) and \(t^*(G) = b\) if and only if \(a \leq b \leq \left\lfloor \frac{3n}{2} \right\rfloor\).
Let \(G\) be a simple graph with \(n\) vertices and \(m\) edges, and let \(\lambda_1\) and \(\lambda_2\) denote the largest and second largest eigenvalues of \(G\). For a nontrivial bipartite graph \(G\), we prove that:
(i) \(\lambda_1 \leq \sqrt{m – \frac{3-\sqrt{5}}{2}}\), where equality holds if and only if \(G \cong P_4\);
(ii) If \(G \ncong P_n\), then \(\lambda_1 \leq \sqrt{{m} – (\frac{5-\sqrt{17}}{2})}\), where equality holds if and only if \(G \cong K_{3,3} – e\);
(iii) If \(G\) is connected, then \(\lambda_2 \leq \sqrt{{m} – 4{\cos}^2(\frac{\pi}{n+1})}\), where equality holds if and only if \(G \cong P_{n,2} \leq n \leq 5\);
(iv) \(\lambda_2 \geq \frac{\sqrt{5}-1}{2}\), where equality holds if and only if \(G \cong P_4\);
(v) If \(G\) is connected and \(G \ncong P_n\), then \(\lambda_2 \geq \frac{5-\sqrt{17}}{2}\), where equality holds if and only if \(G \cong K_{3,3} – e\).
Let \(n\) be a positive integer. Denote by \(PG(n,q)\) the \(n\)-dimensional projective space over the finite field \(\mathbb{F}_q\) of order \(q\). A blocking set in \(PG(n,q)\) is a set of points that has non-empty intersection with every hyperplane of \(PG(n,q)\). A blocking set is called minimal if none of its proper subsets are blocking sets. In this note, we prove that if \(PG(n_i,q)\) contains a minimal blocking set of size \(k_i\) for \(i \in \{1,2\}\), then \(PG(n_1 + n_2 + 1,q)\) contains a minimal blocking set of size \(k_1 + k_2 – 1\). This result is proved by a result on groups with maximal irredundant covers.
A graph is said to be edge-transitive if its automorphism group acts transitively on its edge set. In this paper, all connected cubic edge-transitive graphs of order \(12p\) or \(12p^2\) are classified.
For any \(n\geq 7\), we prove that there exists a tournament of order \(n\), such that for each pair of distinct vertices there exists a path of length \(2\).
A \((k, t)\)-list assignment \(L\) of a graph \(G\) assigns a list of \(k\) colors available at each vertex \(v\) in \(G\) and \(|\bigcup_{v\in V(G)}L(v)| = t\). An \(L\)-coloring is a proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\). A graph \(G\) is \((k,t)\)-choosable if \(G\) has an \(L\)-coloring for every \((k, t)\)-list assignment \(L\).
Erdős, Rubin, and Taylor proved that a graph is \((2, t)\)-choosable for any \(t > 2\) if and only if a graph does not contain some certain subgraphs. Chareonpanitseri, Punnim, and Uiyyasathian proved that an \(n\)-vertex graph is \((2,t)\)-choosable for \(2n – 6 \leq t \leq 2n – 4\) if and only if it is triangle-free. Furthermore, they proved that a triangle-free graph with \(n\) vertices is \((2, 2n – 7)\)-choosable if and only if it does not contain \(K_{3,3} – e\) where \(e\) is an edge. Nakprasit and Ruksasakchai proved that an \(n\)-vertex graph \(G\) that does not contain \(C_5 \vee K_{n-2}\) and \(K_{4,4}\) for \(k \geq 3\) is \((k, kn – k^2 – 2k)\)-choosable. For a non-2-choosable graph \(G\), we find the minimum \(t_1 \geq 2\) and the maximum \(t_2\) such that the graph \(G\) is not \((2, t_i)\)-choosable for \(i = 1, 2\) in terms of certain subgraphs. The results can be applied to characterize \((2, t)\)-choosable graphs for any \(t\).
Let \(G\) be the circuit graph of any connected matroid. It is proved that the circuit graph of a connected matroid with at least three circuits is \(E_2\)-Hamiltonian.
The Randić index \(R(G)\) of a graph \(G\) is defined by \(R(G) = \sum\limits_{uv} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) is the degree of a vertex \(u\) in \(G\) and the summation extends over all edges \(uv\) of \(G\). In this work, we give sharp lower bounds of \(R(G) + g(G)\) and \(R(G) . g(G)\) among \(n\)-vertex connected triangle-free graphs with Randić index \(R\) and girth \(g\).
Hammack and Livesay introduced a new graph operation \(G^{(k)}\) for a graph \(G\), which they called the \(k\)th inner power of \(G\). A graph \(G\) is Hamiltonian if it contains a spanning cycle. In this paper, we show that \(C^{(k)}_n(n \geq 3, k \geq 2)\) is Hamiltonian if and only if \(n\) is odd and \(k = 2\), where \(C_n\) is the cycle with \(n\) vertices.
Let \(a(v)\) and \(g(v)\) denote the least possible area and the least possible number of lattice points in the interior of a convex lattice \(v\)-gon, respectively. Many lower and upper bounds for \(a(v)\) and \(g(v)\) are known for every \(v\). However, the exact values of these two functions are only known for \(v \leq 10\) and \(v \in \{12, 13, 14, 16, 18, 20, 22\}\). The purpose of this paper is to answer the following Open Question 1 from \([13]\): What is the exact value of \(a(11)\)? We answer this question by proving that \(a(11) = 21.5\). On our way to achieve this goal, we also prove that \(g(11) = 17\).
The edge-face total chromatic number of \(3\)-regular Halin graphs was shown to be \(4\) or \(5\) in \([5]\). In this paper, we shall provide a necessary and sufficient condition to characterize \(3\)-regular Halin graphs with edge-face total chromatic number equal to four.
Jaeger \(et \;al\). [ J. Combin. Theory, Ser B, \(56 (1992) 165-182]\) conjectured that every 3-edge-connected graph is \(Z_5\)-connected. Let \(G\) be a 3-edge-connected simple graph on \(n\) vertices and \(A\) an abelian group with \(|A| \geq 3\). If a graph \(G^*\) is obtained by repeatedly contracting nontrivial \(A\)-connected subgraphs of \(G\) until no such subgraph is left, we say \(G\) can be \(A\)-reduced to \(G^*\). It is proved in this paper that \(G\) is \(A\)-connected with \(|A| \geq 5\) if one of the following holds: (i) \(n \leq 15\); (ii) \(n = 16\) and \(\Delta \geq 4\); or (iii) \(n = 17\) and \(\Delta \geq 5\). As applications, we also show the following results:
(1) For \(|A| \geq 5\) and \(n \geq 17\), if \(|E(G)| \geq \binom{n-15}{2} + 31\), then \(G\) is \(A\)-connected.
(2) For \(|A| \geq 4\) and \(n \geq 13\), if \(|E(G)| \geq \binom{n-11}{2} + 23\), then either \(G\) is \(A\)-connected or \(G\) can be \(A\)-reduced to the Petersen graph.
Given a partial cube \(G\), the \(\Theta\)-graph of \(G\) has \(\Theta\)-classes of \(G\) as its vertices, and two vertices in it are adjacent if the corresponding \(\Theta\)-classes meet in a vertex of \(G\). We present a counter-example to the question from \([8]\) whether \(\Theta\)-graphs of graphs of acyclic cubical complexes are always dually chordal graphs. On a positive side, we show that in the class of ACC \(p\)-expansion graphs, each \(\Theta\)-graph is both a dually chordal and a chordal graph. In the proof, a fundamental characterization of \(\Theta\)-acyclic hypergraphs is combined with techniques from metric graph theory. Along the way, we also introduce a new, weaker version of simplicial elimination scheme, which yields yet another characterization of chordal graphs.
Let \(X = (V, E)\) be a connected vertex-transitive graph with degree \(k\). Call \(X\) super restricted edge-connected, in short, sup-\(\lambda’\), if \(F\) is a minimum edge set of \(X\) such that \(X – F\) is disconnected and every component of \(X – F\) has at least two vertices, then \(F\) is the set of edges adjacent to a certain edge in \(X\). Wang [Y, Q, Wang, Super restricted edge-connectivity of vertex-transitive graphs, Discrete Mathematics \(289 (2004) 199-205]\) proved that a connected vertex-transitive graph with degree \(k > 2\) and girth \(g > 4\) is sup-\(\lambda’\). In this paper, by studying the \(k\)-superatom of \(X\), we present sufficient and necessary conditions for connected vertex-transitive graphs and Cayley graphs with degree \(k > 2\) to be sup-\(\lambda’\). In particular, sup-\(\lambda’\) connected vertex-transitive graphs with degree \(k > 2\) and girth \(g > 3\) are completely characterized. These results can be seen as an improvement of the one obtained by Wang.
A proper vertex coloring of a graph \(G\) is called a dynamic coloring if for every vertex \(v\) with degree at least 2, the neighbors of \(v\) receive at least two different colors. It was conjectured that if \(G\) is a regular graph, then \(\chi_2(G) – \chi(G) \leq 2\). In this paper, we prove that, apart from the cycles \(C_4\) and \(C_5\) and the complete bipartite graphs \(K_{n,n}\), every strongly regular graph \(G\) satisfies \(\chi_2(G) – \chi(G) \leq 1\).
Let \(\vec{P_l}\) be the directed path on \(r\) vertices and \(\lambda K^*_{m,n}\) be the symmetric complete bipartite multi-digraph with two partite sets having \(m\) and \(n\) vertices. A \(\vec{P_l}\)-factorization of \(\lambda K^*_{m,n}\) is a set of arc-disjoint \(\vec{P_l}\)-factors of \(\lambda K^*_{m,n}\), which is a partition of the set of arcs of \(\lambda K^*_{m,n}\). In this paper, it is shown that a necessary and sufficient condition for the existence of a \(\vec{P}_{2k+l}\)-factorization of \(\lambda K^*_{m,n}\) for any positive integer \(k\).
Let \(G = (V, E)\) be a finite non-empty graph. A vertex-magic total labeling (VMTL) is a bijection \(\lambda\) from \(V \cup E\) to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\) with the property that for every \(v \in V\), \(\lambda(v) + \sum_{w \in N(v)} \lambda(vw) = h\), for some constant \(h\). Such a labeling is called super if the vertex labels are \(1, 2, \ldots, |V|\).
There are some results known about super VMTLs of \(kG\) only when the graph \(G\) has a super VMTL. In this paper, we focus on the case when \(G\) is the complete graph \(K_n\). It was shown that a super VMTL of \(kK_n\) exists for \(n\) odd and any \(k\), for \(4 < n \equiv 0 \pmod{4}\) and any \(k\), and for \(n = 4\) and \(k\) even. We continue the study and examine the graph \(kK_n\) for \(n \equiv 2 \pmod{4}\). Let \(n = 4l + 2\) for a positive integer \(l\). The graph \(kK_{4l+2}\) does not admit a super VMTL for \(k\) odd. We give a large number of super VMTLs of \(kK_{4l+2}\) for any even \(k\) based on super VMTLs of \(4K_{2l+1}\).
For a given graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph. Let \(K_m – H\) be the graph obtained from \(K_m\) by removing the edge set \(E(H)\), where \(H\) is a subgraph of \(K_m\). In this paper, we characterize the potentially \(K_6 – C_4\)-graphic sequences. This characterization implies a theorem due to Hu and Lai \([7]\).
Double Fibonacci sequences \((x_{n,k})\) are introduced and they are related to operations with Fibonacci modules. Generalizations and examples are also discussed.
A set \(S \subseteq V\) is a dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is either in \(S\) or is adjacent to a vertex in \(S\). A vertex is said to dominate itself and all its neighbors. The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). In terms of a chess board problem, let \(X_n\) be the graph for chess piece \(X\) on the square of side \(n\). Thus, \(\gamma(X_n)\) is the domination number for chess piece \(X\) on the square of side \(n\). In 1964, Yaglom and Yaglom established that \(\gamma(K_n) = \left\lceil \frac{n+2}{2} \right\rceil^2\). This extends to \(\gamma(K_{m,n}) = \left\lceil \frac{m+2}{3} \right\rceil \left\lceil \frac{n+2}{3} \right\rceil\) for the rectangular board. A set \(S \subseteq V\) is a total dominating set of a graph \(G = (V, E)\) if each vertex in \(V\) is adjacent to a vertex in \(S\). A vertex is said to dominate its neighbors but not itself. The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king’s graph on the \(m \times n\) board. In this paper, we note an error in one portion of their analysis and provide a correct general upper bound for \(\gamma_t(K_{m,n})\). Furthermore, we state improved upper bounds for \(\gamma_t(K_n)\).
A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually the positive integers). An \((a, d)\)-edge-antimagic total labeling of a graph with \(p\) vertices and \(q\) edges is a one-to-one mapping that takes the vertices and edges onto the integers \(1, 2, \ldots, p + q\), such that the sums of the label on the edges and the labels of their end points form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called \({super}\) if the smallest possible labels appear on the vertices. In this paper, we study the super \((a, 2)\)-edge-antimagic total labelings of disconnected graphs. We also present some necessary conditions for the existence of \((a, d)\)-edge-antimagic total labelings for \(d\) even.
Fault tolerance is an important property of network performance. A graph \(G\) is \(k\)-edge-fault conditional Hamiltonian if \(G – F\) is Hamiltonian for every \(F \subset E(G)\) with \(|F| \leq k\) and \(\delta(G – F) \geq 2\). In this paper, we show that for \(n \geq 4\), the \(n\)-dimensional star graph \(S_n\) is \((3n – 10)\)-edge-fault conditional Hamiltonian.
In this paper, we characterize all spacelike, timelike, and null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\), in Minkowski space \({E}^5_v\). Moreover, we prove that there are no timelike and no null curves lying on the pseudohyperbolic space \({H}^{4}_{v-1}\) in \({E}^5_v\).
The local-restricted-edge-connectivity \(\lambda'(e, f)\) of two nonadjacent edges \(e\) and \(f\) in a graph \(G\) is the maximum number of edge-disjoint \(e\)-\(f\) paths in \(G\). It is clear that \(\lambda'(G) = \min\{\lambda'(e, f) \mid e \text{ and } f \text{ are nonadjacent edges in } G\}\), and \(\lambda'(e, f) \leq \min\{\xi(e), \xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\), where \(\lambda(G)\), \(\xi(e)\), and \(\xi(f)\) denote the restricted-edge-connectivity of \(G\), the edge-degree of edges \(e\) and \(f\), respectively. Let \(\xi(G)\) be the minimum edge-degree of \(G\). We call a graph \(G\) optimally restricted-edge-connected when \(\lambda'(G) = \xi(G)\) and optimally local-restricted-edge-connected if \(\lambda'(e, f) = \min\{\xi(e),\xi(f)\}\) for all pairs \(e\) and \(f\) of nonadjacent edges in \(G\). In this paper, we show that some known sufficient conditions that guarantee that a graph is optimally restricted-edge-connected also guarantee that it is optimally local-restricted-edge-connected.
In 1982, Beutelspacher and Brestovansky proved that for every integer \(m \geq 3\), the \(2\)-color Rado number of the equation
\[x_1+x_2+ \ldots + x_{m-1}=x_m\]
is \(m^2 – m – 1\). In 2008, Schaal and Vestal proved that, for every \(m \geq 6\), the \(2\)-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=2x_m\]
is \(\left\lceil \frac{m-1}{2}\left\lceil \frac{m-1}{2} \right\rceil \right\rceil \). Here, we prove that, for every integer \(a \geq 3\) and every \(m \geq 2a^2 – a + 2\), the 2-color Rado number of
\[x_1+x_2+ \ldots + x_{m-1}=ax_m\]
is \(\left\lceil \frac{m-1}{a}\left\lceil \frac{m-1}{a} \right\rceil \right\rceil\). For the case \(a = 3\), we show that our formula gives the Rado number for all \(m \geq 7\), and we determine the Rado number for all \(m \geq 3\).
The general Randic index \(R_{-\alpha}(G)\) of a graph \(G\), defined by a real number \(\alpha\), is the sum of \((d(u)d(v))^{-\alpha}\) over all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we have discussed some properties of the Max Tree which has the maximum general Randic index \(R_{-\alpha}(G)\), where \(\alpha \in (\alpha_0,2)\). Based on these properties, we are able to obtain the structure of the Max Tree among all trees of order \(k \geq 3\). Thus, the maximal value of \(R_{-\alpha}(G)\) follows easily.
A \(\lambda\)-fold \(G\)-design of order \(n\) is a pair \((X, {B})\), where \(X\) is a set of \(n\) vertices and \({B}\) is a collection of edge-disjoint copies of the simple graph \(G\), called blocks, which partitions the edge set of \(K_n\) (the undirected complete graph with \(n\) vertices) with vertex set \(X\). Let \((X, {B})\) be a \(G\)-design and \(H\) be a subgraph of \(G\). For each block \(B \in \mathcal{B}\), partition \(B\) into copies of \(H\) and \(G \setminus H\) and place the copy of \(H\) in \({B}(H)\) and the edges belonging to the copy of \(G \setminus H\) in \({D}(G \setminus H)\). Now, if the edges belonging to \({D}(G \setminus H)\) can be arranged into a collection \({D}_H\) of copies of \(H\), then \((X, {B}(H) \cup {D}(H))\) is a \(\lambda\)-fold \(H\)-design of order \(n\) and is called a metamorphosis of the \(\lambda\)-fold \(G\)-design \((X, {B})\) into a \(\lambda\)-fold \(H\)-design, denoted by \((G > H) – M_\lambda(n)\).
In this paper, the existence of a \((G > H) – M_\lambda(n)\) for graph designs will be presented, variations of this problem will be explained, and recent developments will be surveyed.
For an integer \(k \geq 1\) and a graph \(G = (V, E)\), a subset \(S\) of the vertex set \(V\) is \(k\)-independent in \(G\) if the maximum degree of the subgraph induced by the vertices of \(S\) is less than or equal to \(k – 1\). The \(k\)-independence number \(\beta_k(G)\) of \(G\) is the maximum cardinality of a \(k\)-independent set of \(G\). A set \(S\) of \(V\) is \(k\)-Co-independent in \(G\) if \(S\) is \(k\)-independent in the complement of \(G\). The \(k\)-Co-independence number \(\omega_k(G)\) of \(G\) is the maximum size of a \(k\)-Co-independent set in \(G\). The sequences \((\beta_k)\) and \((\omega_k)\) are weakly increasing. We define the \(k\)-chromatic number or \(k\)-independence partition number \(\chi_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-independent sets and the \(k\)-Co-independence partition number \(\theta_k(G)\) of \(G\) as the smallest integer \(m\) such that \(G\) admits a partition of its vertices into \(m\) \(k\)-Co-independent sets. The sequences \((\chi_k)\) and \((\theta_k)\) are weakly decreasing. In this paper, we mainly present bounds on these four parameters, some of which are extensions of well-known classical results.
It is proved that if \(G\) is a plane embedding of a \(K_4\)-minor-free graph, then \(G\) is coupled \(5\)-choosable; that is, if every vertex and every face of \(G\) is given a list of \(5\) colours, then each of these ele-ments can be given a colour from its list such that no two adjacent or incident elements are given the same colour. Using this result it is proved also that if \(G\) is a plane embedding of a \(K_{2,3}\),\(3\)-minor-free graph or a \((\bar{K}_2 + (K_1 \cup K_2))\)-minor-free graph, then \(G\) is coupled \(5\)-choosable. All results here are sharp, even for outerplane graphs.
A Steiner system \(S(2, k, v)\) is a collection of \(k\)-subsets (blocks) of a \(k\)-set \(V\) such that each \(2\)-subset of \(V\) is contained in exactly one block. We find re-currence relations for \(S(2, k, v)\).
Denote by \(\mathcal{P}(n_1, n_2, n_3)\) the set of all polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\). Let \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)) be three non-isomorphic polyphenyl spiders with three legs of lengths \(n_1\), \(n_2\), and \(n_3\), and let \(m_k(G)\) and \(i_k(G)\) be the numbers of \(k\)-matchings and \(k\)-independent sets of a graph \(G\), respectively. In this paper, we show that for any \(S^j(n_1, n_2, n_3) \in \mathcal{P}(n_1, n_2, n_3)\) (\(j \in \{1, 2, 3\}\)), we have \(m_k(S_M^3(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3)) \leq m_k(S^j(n_1, n_2, n_3))\) and \(i_k(S_O^1(n_1, n_2, n_3)) \leq i_k(S^j(n_1, n_2, n_3)) \leq i_k(S^3_M(n_1, n_2, n_3))\), with equalities if and only if \(S^j(n_1, n_2, n_3) = S_M^3(n_1, n_2, n_3)\) or \(S^j(n_1, n_2, n_3) = S_O^1(n_1, n_2, n_3)\), where \(S_O^1(n_1, n_2, n_3)\) and \(S_M^3(n_1, n_2, n_3)\) are respectively an ortho-polyphenyl spider and a meta-polyphenyl spider.
A set \( S \) of vertices in a graph \( G \) is a total dominating set of \( G \) if every vertex of \( G \) is adjacent to some vertex in \( S \). The minimum cardinality of a total dominating set of \( G \) is the total domination number of \( G \). We study graphs having the same total domination number as their complements. In particular, we characterize the cubic graphs having this property. Also, we characterize such graphs with total domination numbers equal to two or three, and we determine properties of the ones with larger total domination numbers.
In 1991 Gnanajothi conjectured: Each tree is odd-graceful. In this paper, we define the edge-ordered odd-graceful labeling of trees and show the odd-gracefulness of all symmetric trees.
A method is suggested for the construction of quadrangulations of the closed orientable surface with given genus \( g \) and either (1) with a given chromatic number or (2) with a given order allowed by the genus \( g \). In particular, N. Hartshfield and G. Ringel’s results [J. Comb. Theory, Ser. B 46 (1989), 84-95] are generalized by way of generating minimal quadrangulations of infinitely many other genera.
For integers \( s, t \geq 1 \), the Ramsey number \( R(s, t) \) is defined to be the least positive integer \( n \) such that every graph on \( n \) vertices contains either a clique of order \( s \) or an independent set of order \( t \). In this note, the lower bound for the Ramsey number \( R(7, 9) \) is improved from \( 241 \) to \( 242 \). The new bound is obtained by searching the maximum common induced subgraph between two graphs with a depth variable local search technique.
In this paper, we give an alternative and more intuitive proof to one of two classic inequalities given by Diaconis and Graham in 1977. The inequality involves three metrics on the symmetric group, i.e., the set of all permutations of the first \( n \) positive integers. Our technique for the proof of the inequality allows us to resolve an open problem posed in that paper: When does equality hold? It also allows us to estimate how often equality holds. In addition, our technique can sometimes be applied for the proof of other inequalities between metrics or pseudo-metrics on the symmetric group.
Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we construct a family of error-correcting pooling designs with the incidence matrix of two types of subspaces of singular linear space over finite fields, and exhibit their disjunct properties.
Let \( S \) be an orthogonal polygon in the plane, bounded by a simple closed curve, and let \( R \) be the smallest rectangular region containing \( S \). Assume that \( S \) is star-shaped via staircase paths. For every point \( p \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \), there is a corresponding point \( q \) in \( \text{bdry} \, S \) such that \( p \) lies in a maximal staircase convex cone \( C_q \) at \( q \) in \( \mathbb{R}^2 \setminus (\text{int} \, S) \). Furthermore, point \( q \) may be selected to satisfy these requirements:
Thus we obtain a finite family of staircase convex cones whose union is \( \mathbb{R}^2 \setminus (\text{int} \, S) \).
If there are integers \( k \) and \( \lambda \neq 0 \) such that a total labeling \( f \) of a connected graph \( G = (V, E) \) from \( V \cup E \) to \( \{1, 2, \ldots, |V| + |E|\} \) satisfies \( f(x) \neq f(y) \) for distinct \( x, y \in V \cup E \) and
\[ f(u) + f(v) = k + \lambda f(uv) \]
for each edge \( uv \in E \), then \( f \) is called a \( (k, \lambda) \)-\({magically\; total\; labeling}\) (\( (k, \lambda) \)-\({mtl}\) for short) of \( G \). Several properties of \( (k, \lambda) \)-\({mtls}\) of graphs are shown. The sufficient and necessary connections between \( (k, \lambda) \)-\emph{mtls} and several known labelings (such as graceful, odd-graceful, felicitous, and \( (b, d) \)-edge antimagic total labelings) are given. Furthermore, every tree is proven to be a subgraph of a tree having super \( (k, \lambda) \)-\({mtls}\).
Let \( G \) be a simple graph of order \( n \), and let \( k \) be a positive integer. A graph \( G \) is fractional independent-set-deletable \( k \)-factor-critical (in short, fractional ID-\( k \)-factor-critical) if \( G – I \) has a fractional \( k \)-factor for every independent set \( I \) of \( G \). In this paper, we obtain a sufficient condition for a graph \( G \) to be fractional ID-\( k \)-factor-critical. Furthermore, it is shown that the result in this paper is best possible in some sense.
Calculations of the number of equivalence classes of Sudoku boards has to this point been done only with the aid of a computer, in part because of the unnecessarily large symmetry group used to form the classes. In particular, the relationship between relabeling symmetries and positional symmetries such as row/column swaps is complicated. In this paper, we focus first on the smaller Shidoku case and show first by computation and then by using connectivity properties of simple graphs that the usual symmetry group can in fact be reduced to various minimal subgroups that induce the same action. This is the first step in finding a similar reduction in the larger Sudoku case and for other variants of Sudoku.
Let \( k \) be a positive integer, and let \( G \) be a simple graph with vertex set \( V(G) \). A function \( f: V(G) \to \{\pm1, \pm2, \ldots, \pm k\} \) is called a signed \(\{k\}\)-dominating function if
\[ \sum_{u \in N[v]} f(u) \geq k \]
for each vertex \( v \in V(G) \).
The signed \(\{1\}\)-dominating function is the same as the ordinary signed domination. A set \( \{f_1, f_2, \ldots, f_d\} \) of signed \(\{k\}\)-dominating functions on \( G \) with the property that
\[ \sum_{i=1}^d f_i(v) \leq k \]
for each \( v \in V(G) \), is called a \({signed \;\{k\}-dominating \;family}\) (of functions) on \( G \). The maximum number of functions in a signed \(\{k\}\)-dominating family on \( G \) is the \({signed \;\{k\}-domatic\; number}\) of \( G \), denoted by \( d_{\{k\}S}(G) \). Note that \( d_{\{1\}S}(G) \) is the classical signed domatic number \( d_s(G) \).
In this paper, we initiate the study of signed \(\{k\}\)-domatic numbers in graphs, and we present some sharp upper bounds for \( d_{\{k\}S}(G) \). In addition, we determine \( d_{\{k\}S}(G) \) for several classes of graphs. Some of our results are extensions of known properties of the signed domatic number.
A graceful \( n \)-permutation is a graceful labeling of an \( n \)-vertex path \( P_n \). In this paper, we improve the asymptotic lower bound on the number of such permutations from \( \Omega\left(\left(\frac{5}{3}\right)^n\right) \) to \( \Omega\left(2.37^n\right) \). This is a computer-assisted proof based on an effective algorithm that enumerates graceful \( n \)-permutations. Our algorithm is also presented in detail.
Let \( K_{r+1} \) be the complete graph on \( r+1 \) vertices and let \( \pi = (d_1, d_2, \ldots, d_n) \) be a non-increasing sequence of nonnegative integers. If \( \pi \) has a realization containing \( K_{r+1} \) as a subgraph, then \( \pi \) is said to be potentially \( K_{r+1} \)-graphic. A.R. Rao obtained an Erdős-Gallai type criterion for \( \pi \) to be potentially \( K_{r+1} \)-graphic. In this paper, we provide a simplification of this Erdős-Gallai type criterion. Additionally, we present the Fulkerson-Hoffman-McAndrew type criterion and the Hasselbarth type criterion for \( \pi \) to be potentially \( K_{r+1} \)-graphic.
In this paper, we introduce several concepts related to fuzzy algebraic structures. We provide an example of a fuzzy binary operation and a fuzzy group. Additionally, we define a new fuzzy binary operation on a \(\Gamma\)-ring \(M\) and introduce a new fuzzy \(\Gamma\)-ring. We also present homomorphism theorems between two fuzzy \(\Gamma\)-rings and investigate some related properties.
Let \( R \) be a commutative ring and \( Z(R) \) be its set of all zero-divisors. The \emph{total graph} of \( R \), denoted by \( T_\Gamma(R) \), is the undirected graph with vertex set \( R \), where two distinct vertices \( x \) and \( y \) are adjacent if and only if \( x + y \in Z(R) \).
In this paper, we obtain a lower bound as well as an upper bound for the domination number of \( T_\Gamma(R) \). Further, we prove that the upper bound for the domination number of \( T_\Gamma(R) \) is attained in the case of an Artin ring \( R \). Having established this, we identify certain classes of rings for which the domination number of the total graph equals this upper bound.
In view of these results, we conjecture that the domination number of \( T_\Gamma(R) \) is always equal to this upper bound. We also derive certain other domination parameters for \( T_\Gamma(R) \) under the assumption that the conjecture is true.
For given graphs \( H_1 \) and \( H_2 \), the \({Ramsey\; number}\) \( R(H_1, H_2) \) is the smallest positive integer \( n \) such that if we arbitrarily color the edges of the complete graph \( K_n \) with two colors, 1 (red) and 2 (blue), then there is a monochromatic copy of \( H_1 \) colored with 1 or \( H_2 \) colored with 2.
We show that if \( n \) is even, \( q = \lceil \sqrt{n} \rceil \) is odd, and \( s = n – (q-1)^2 \leq \frac{q}{2} \), then \( R(K_{2,2}, K_{2,n}) \leq n + 2q – 1 \), where \( K_{n,m} \) are complete bipartite graphs. This bound provides the exact value of \( R(K_{2,2}, K_{2,18}) = 27 \). Moreover, we show that \( R(K_{2,2}, K_{2,14}) = 22 \) and \( R(K_{2,2}, K_{2,15}) = 24 \).
A \({red-blue\; coloring}\) of a graph \( G \) is an edge coloring of \( G \) in which every edge is colored red or blue. For a connected graph \( H \) of size at least 2, a \({color \;frame}\) \( F \) of \( H \) is obtained from a red-blue coloring of \( H \) having at least one edge of each color and in which a blue edge is designated as the root edge.
An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \), and the \( F \)-\({chromatic\; index}\) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). An \( F \)-coloring of \( G \) is \({minimal}\) if whenever any red edge of \( G \) is changed to blue, then the resulting red-blue coloring of \( G \) is not an \( F \)-coloring of \( G \). The maximum number of red edges in a minimal \( F \)-coloring of \( G \) is the \({upper \; F -chromatic \;index}\) of \( G \).
In this paper, we investigate \( F \)-colorings and \( F \)-chromatic indexes of graphs for all color frames \( F \) of paths of orders 3 and 4.
A set of vertices \( S \) in a graph \( G \) is a dominating set if any vertex of \( G – S \) is adjacent to some vertex in \( S \). The domination number, \( \gamma(G) \), of \( G \) is the minimum cardinality of a dominating set of \( G \).
The subdivision of an edge \( uv \) is the operation of replacing \( uv \) with a path \( uwv \) through a new vertex \( w \). A graph \( G \) is domination critical upon edge subdivision if the domination number increases by subdivision of any edge.
In this paper, we study domination critical graphs upon edge subdivision. We present several properties and bounds for these graphs and then give a constructive characterization of domination critical trees upon edge subdivision.
Let \( G \) be a graph of order \( n \). The \({binding\; number}\) of \( G \) is defined as
\[
\text{bind}(G) := \min \left\{ \frac{|N_G(X)|}{|X|} \mid \emptyset \neq X \subseteq V(G) \text{ and } N_G(X) \neq V(G) \right\}.
\]
A \((g, f)\)-factor is called a connected \((g, f)\)-factor if it is connected. A \((g, f)\)-factor \( F \) is called a Hamilton \((g, f)\)-factor if \( F \) contains a Hamilton cycle. In this paper, several sufficient conditions related to binding number and minimum degree for graphs to have connected \((g, f+1)\)-factors or Hamilton \((g, f)\)-factors are given.
Define an edge \( Q_1Q_2 \) or a triangle \( Q_1Q_2Q_3 \) of a clique graph \( K(G) \) to be weight-\( k \) if \( |Q_1 \cap Q_2| \geq k \) or \( |Q_1 \cap Q_2 \cap Q_3| \geq k \), respectively. A graph \( G \) is shown to be strongly chordal if and only if, for every \( k \geq 1 \), every cycle of weight-\( k \) edges in \( K(G) \) either has a weight-\( k \) chord or is a weight-\( k \) triangle—this mimics the usual definition of chordal graphs. Similarly, trivially perfect graphs have a characterization that mimics a simple characterization of component-complete graphs.
We propose an original approach to the problem of rank-unimodality for Dyck lattices. It is based on a well-known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of the problem of unimodality of succession rules.
A Roman dominating function on a graph \( G \) is a labeling \( f: V(G) \to \{0, 1, 2\} \) such that every vertex with label \( 0 \) has a neighbor with label \( 2 \). The weight of a Roman dominating function is the value \( f(V(G)) = \sum_{u \in V(G)} f(u) \). The minimum weight of a Roman dominating function on a graph \( G \) is called the Roman domination number, denoted by \( \gamma_R(G) \). The Roman bondage number of a graph \( G \) is the cardinality of a smallest set of edges whose removal results in a graph with Roman domination number greater than that of \( G \).
In this paper, we initiate the study of the Roman fractional bondage number, and we present different bounds on Roman fractional bondage. In addition, we determine the Roman fractional bondage number of some classes of graphs.
We show that the principal results of the article “The metric dimension of graphs with pendant edges” [Journal of Combinatorial Mathematics and Combinatorial Computing, 65 (2008) 139-145] do not hold. In this paper, we correct the results and we solve two open problems described in the above-mentioned paper.
Using the definition of the representation number of a graph modulo integers given by Erdős and Evans, we establish the representation number of a complete graph minus a set of disjoint stars. The representation number of a graph \( G \) is the smallest positive integer \( n \) for which there is a labeling of every vertex of \( G \) with a distinct element of \( \{0,1,2,\ldots,n-1\} \) such that two vertices are adjacent if and only if the difference of their labels is relatively prime to \( n \). We apply known results to a complete graph minus a set of stars to establish a lower bound for the representation number; then show a systematic labeling of the vertices producing a representation that attains that lower bound. Thus showing that for complete graphs minus a set of disjoint stars, the established lower bound of the representation number modulo \( n \) is indeed the representation number of the graph. Since the representation modulo an integer for a complete graph minus disjoint stars is attained using the fewest number of primes allowed by the lower bound, it follows that the corresponding Prague dimension will be determined by the largest star removed from the complete graph.
Let \(\lambda K_v\) be the complete multigraph of order \(v\) and index \(\lambda\), where any two distinct vertices \(x\) and \(y\) are joined exactly by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design of \(\lambda K_v\), denoted by \((v,G,\lambda)\)-GD, is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(\lambda K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly \(\lambda\) blocks of \(\mathcal{B}\). There are four graphs which are a 6-circle with two pendant edges, denoted by \(G_i\), \(i = 1,2,3,4\). In [9], we have solved the existence problems of \((v, G_i, 1)\)-GD. In this paper, we obtain the existence spectrum of \((v, G_i, \lambda)\)-GD for any \(\lambda > 1\).
The decycling index of a digraph is the minimum number of arcs whose removal yields an acyclic digraph. The maximum arc decycling number \(\overline{\nabla}'(m,n)\) is the maximum decycling index among all \(m\times n\) bipartite tournaments. Recently, R.C. Vandell determined the numbers \(\overline{\nabla}'(2,n)\), \(\overline{\nabla}'(3,n)\), and \(\overline{\nabla}'(4,n)\) for all positive integers \(n\), as well as \(\overline{\nabla}'(5,5)\). In this work, we use a computer program to obtain \(\overline{\nabla}'(5,6)\), \(\overline{\nabla}'(6,6)\), and \(\overline{\nabla}'(5,7)\), as well as some results on \(\overline{\nabla}'(6,7)\) and \(\overline{\nabla}'(5,8)\). In particular, \(\overline{\nabla}'(6,6) = 10\), and this confirms a conjecture of Vandell.
Let \( G = (V, E) \) be a graph. A function \( f: V \to \{-1, 1\} \) is called a signed dominating function on \( G \) if \( \sum_{u \in N_G[v]} f(u) \geq 1 \) for each \( v \in V \), where \( N_G[v] \) is the closed neighborhood of \( v \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed dominating functions on \( G \) is called a signed dominating family (of functions) on \( G \) if \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V \). The signed domatic number of \( G \) is the maximum number of functions in a signed dominating family on \( G \). The signed total domatic number is defined similarly, by replacing the closed neighborhood \( N_G[v] \) with the open neighborhood \( N_G(v) \) in the definition. In this paper, we prove that the problems of computing the signed domatic number and the signed total domatic number of a given graph are both NP-hard, even if the graph has bounded maximum degree. To the best of our knowledge, these are the first NP-hardness results for these two variants of the domatic number.
Consider the following problem: Given a transitive tournament \(T\) of order \(n \geq 3\) and an integer \(k\) with \(1 \leq k \leq \binom{n}{2}\), which \(k\) ares in \(T\) should be reversed so that the resulting tournament has the largest number of spanning cycles? In this note, we solve the problem when \(7\) is sufficiently large compared to \(k\).
The bondage number \(b(G)\) of a graph \(G\) is the smallest number of edges whose removal results in a graph with domination number greater than the domination number of \(G\). Kang and Yuan [Bondage number of planar graphs. Discrete Math. \(222 (2000), 191-198]\) proved \(b(G) \leq \min\{8, \Delta + 2\}\) for every connected planar graph \(G\), where \(\Delta\) is the maximum degree of \(G\). Later Carlson and Develin [On the bondage number of planar and directed graphs. Discrete Math. \(306 (8-9) (2006), 820-826]\) presented a method to give a short proof for this result. This paper applies this technique to generalize the result of Kang and Yuan to any connected graph with crossing number less than four.
A \({Roman \;domination \;function}\) on a graph \(G = (V, E)\) is a function \(f: V(G) \to \{0, 1, 2\}\) satisfying the condition that every vertex \(u\) with \(f(u) = 0\) is adjacent to at least one vertex \(v\) with \(f(v) = 2\). The \({weight}\) of a Roman domination function \(f\) is the value \(f(V(G)) = \sum_{u \in V(G)} f(u)\). The minimum weight of a Roman dominating function on a graph \(G\) is called the \({Roman \;domination \;number}\) of \(G\), denoted by \(\gamma_R(G)\). In this paper, we study the Roman domination number of generalized Petersen graphs \(P(n, 2)\) and prove that \(\gamma_R(P(n, 2)) = \left\lceil \frac{8n}{7} \right\rceil (n\geq5)\).
Let \(G = (V, E)\) be a simple undirected graph. For an edge \(e\) of \(G\), the \({closed\; edge-neighborhood}\) of \(e\) is the set \(N[e] = \{e’ \in E \mid e’ \text{ is adjacent to } e\} \cup \{e\}\). A function \(f: E \to \{1, -1\}\) is called a signed edge domination function (SEDF) of \(G\) if \(\sum_{e’ \in N[e]} f(e’) > 1\) for every edge \(e\) of \(G\). The signed edge domination number of \(G\) is defined as \(\gamma’_s(G) = \min \left\{ \sum_{e \in E} |f(e)| \mid f \text{ is an SEDF of } G \right\}\). In this paper, we determine the signed edge domination numbers of all complete bipartite graphs \(K_{m,n}\), and therefore determine the signed domination numbers of \(K_m \times K_n\).
We discuss the primality of some corona graphs and some families of graphs.
An injective coloring of a graph \(G\) is an assignment of colors to the vertices of \(G\) so that any two vertices with a common neighbor receive distinct colors. A graph \(G\) is said to be injectively \(k\)-choosable if any list \(L(v)\) of size at least \(k\) for every vertex \(v\) allows an injective coloring \(\phi(v)\) such that \(\phi(v) \in L(v)\) for every \(v \in V(G)\). The least \(k\) for which \(G\) is injectively \(k\)-choosable is the injective choosability number of \(G\), denoted by \(\chi_i^l(G)\). In this paper, we obtain new sufficient conditions to ensure \(\chi_i^l(G) \leq \Delta(G) + 1\). We prove that if \(mad(G) \leq \frac{12k}{4k+3}\), then \(\chi_i^l(G) = \Delta(G) + 1\) where \(k = \Delta(G)\) and \(k \geq 4\). Typically, proofs using the discharging technique are different depending on maximum average degree \(mad(G)\) or maximum degree \(\Delta(G)\). The main objective of this paper is finding a function \(f(\Delta(G))\) such that \(\chi_i^l(G) \leq \Delta(G) + 1\) if \(mad(G) < f(\Delta(G))\), which can be applied to every \(\Delta(G)\).
The traditional parameter used as a measure of vulnerability of a network modeled by a graph with perfect nodes and edges that may fail is edge connectivity \(\lambda\). For the complete bipartite graph \(K_{p,q}\), where \(1 \leq p \leq q\), \(\lambda(K_{p,q}) = p\). In this case, failure of the network means that the surviving subgraph becomes disconnected upon the failure of individual edges. If, instead, failure of the network is defined to mean that the surviving subgraph has no component of order greater than or equal to some preassigned number \(k\), then the associated vulnerability parameter, the component order edge connectivity \(\lambda_c^{(k)}\), is the minimum number of edges required to fail so that the surviving subgraph is in a failure state. We determine the value of \(\lambda_c^{(k)}(K_{p,q})\) for arbitrary \(1 \leq p \leq q\) and \(4 \leq k \leq p+q\). As it happens, the situation is relatively simple when \(p\) is small and more involved when \(p\) is large.
A \(T\)-shape tree \(T(l_1, l_2, l_3)\) is obtained from three paths \(P_{l_1+1}\), \(P_{l_2+1}\), and \(P_{l_3+1}\) by identifying one of their pendent vertices. A generalized \(T\)-shape tree \(T_s(l_1, l_2, l_3)\) is obtained from \(T(l_1, l_2, l_3)\) by appending two pendent vertices to exactly \(s\) pendent vertices of \(T(l_1, l_2, l_3)\), where \(1 \leq s \leq 3\) is a positive integer. In this paper, we firstly show that the generalized \(T\)-shape tree \(T_2(l_1, l_2, l_3)\) is determined by its Laplacian spectrum. Applying similar arguments for the trees \(T_1(2l_1, l_2, l_3)\) and \(T_3(l_1, 2l_2, l_3)\), one can obtain that any generalized \(T\)-shape tree on \(n\) vertices is determined by its Laplacian spectrum.
In this paper, we use the \(q\)-difference operator and the Andrews-Askey integral to give a transformation for the Al-Salam-Carlitz polynomials. As applications, we obtain an expansion of the Carlitz identity and some other identities for Al-Salam-Carlitz
polynomials .
In this paper we define new generalizations of the Lucas numbers,which also generalize the Perrin numbers. This generalization is based on the concept of \(k\)-distance Fibonacci numbers. We give in-terpretations of these numbers with respect to special decompositions and coverings, also in graphs. Moreover, we show some identities for these numbers, which often generalize known classical relations for the Lucas numbers and the Perrin numbers. We give an application of the distance Fibonacci numbers for building the Pascal’s triangle.
This paper introduces the new notions of \(\delta-\alpha-\)open sets and the \(\delta-\alpha-\)continuous functions in the topological spaces and investigates some of their properties.
Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (n_1g) \cdots (n_lg)\), where \(g \in G\) and \(n_1, \ldots, n_l \in [1, \text{ord}(g)]\), and the \({index}\) \(\text{ind}(S)\) of \(S\) is defined to be the minimum of \((n_1 + \cdots + n_l)/\text{ord}(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) when \(G = \langle g \rangle\) is a cyclic group of a prime order and \(S\) has the form \(S = g^2{(n_2g)}(n_3g){(n_4)}\). It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime order \(p \geq 31\), then every minimal zero-sum sequence \(S\) of the above-mentioned form has index \(1\), except in the case that \(S = g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)\).
The paper presents two sharp upper bounds for the largest Laplacian eigenvalue of mixed graphs in terms of the degrees and the average \(2\)-degrees, which improve and generalize the main results of Zhang and Li [Linear Algebra Appl.\(353(2002)11-20]\),Pan (Linear Algebra Appl.\(355(2002)287-295]\),respectively. Moreover, we also characterize some extreme graphs which attain these upper bounds. In last, some examples show that our bounds are improvement on some known bounds in some cases.
Cagman \(et\; al\). introduced the concept of a fuzzy parameterized fuzzy soft set(briefly, \(FPFS)\) which is an extension of a fuzzy set and a soft set. In this paper, we introduce the concepts of \(FPFS\) filters and \(FPFS\) implicative filters of lattice implication algebras and obtain some related results. Finally, we define the concept of \(FPFS\)-aggregation operator of lattice implication algebras.
We propose a practical linear time algorithm for the LONGEST PATH problem on \(2\)-trees.
By means of a \(q\)-binomial identity, we give two generalizations of Prodinger’s formula, which is equivalent to the famous Dilcher’s formula.
In this paper, we consider a random mapping \(\hat{T}_{n,\theta}\) of the finite set \(\{1,2,\ldots,n\}\) into itself, for which the digraph representation \(\hat{G}_{n,\theta}\) is constructed by: (1) selecting a random number \(\hat{L}_n\) of cyclic vertices, (2) constructing a uniform random forest of size \(n\) with the selected cyclic vertices as roots, and (3) forming `cycles’ of trees by applying to the selected cyclic vertices a random permutation with cycle structure given by the Ewens sampling formula with parameter \(\theta\). We investigate \(\hat{k}_{n,\theta}\), the size of a `typical’ component of \(\hat{G}_{n,\theta}\), and we obtain the asymptotic distribution of \(\hat{k}_{n,\theta}\) conditioned on \(\hat{L}_n = m(n)\). As an application of our results, we show in Section 3 that provided \(\hat{L}_n\) is of order much larger than \(\sqrt{n}\), then the joint distribution of the normalized order statistics of the component sizes of \(G_{n,\theta}\) converges to the Poisson-Dirichlet \((\theta)\) distribution as \(n \to \infty\).
In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, D. S. Kim and T. Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus \((see[6])\).
Let \(H\) be a subgraph of \(G\). An \(H\)-design \((V, \mathcal{C})\) of order \(v\) and index \(\lambda\) is embedded into a \(G\)-design \((X, \mathcal{B})\) of order \(v+w\), \(w \geq 0\), and index \(\lambda\), if \(\mu \leq \lambda\), \(V \subseteq X\) and there is an injective mapping \(f: \mathcal{C} \rightarrow \mathcal{B}\) such that \(B\) is a subgraph of \(f(B)\) for every \(B \in \mathcal{C}\).
For every pair of positive integers \(v\) and \(\lambda\), we determine the minimum value of \(w\) such that there exists a balanced incomplete block design of order \(v+w\), index \(\lambda \geq 2\) and block-size \(4\) which embeds a \(K_3\)-design of order \(v\) and index \(\mu = 1\).
Let \(S\) be a finite, nonempty set of nonzero integers which contains no squares. We obtain conditions both necessary and sufficient for \(S\) to have the following property: for infinitely many primes \(p\), \(S\) is a set of quadratic nonresidues of \(p\). The conditions are expressed solely in terms of purely external (respectively, internal) combinatorial properties of the set II of all prime factors of odd multiplicity of the elements of \(S\). We also calculate by means of certain purely combinatorial parameters associated with \(\prod\) the density of the set of all primes \(p\) such that \(S\) is a set of quadratic residues of \(p\) and the density of the set of all primes \(p\) such that \(S\) is a set of quadratic nonresidues of \(p\).
For positive integers \(t\) and \(k\), the \({vertex}\) (resp. edge) Folkman number \(F_v(t,t,t;k)\) (resp. \(F_e(t,t,t;k)\)) is the smallest integer \(n\) such that there is a \(K_k\)-free graph of order \(n\) for which any three coloring of its vertices (resp. edges) yields a monochromatic copy of \(K_t\). In this note, an algorithm for testing \((t,t,\ldots,t;k)\) in cyclic graphs is presented and it is applied to find new upper bounds for some vertex or edge Folkman numbers. By using this method, we obtain \(F_v(3,3,3;4) \leq 66\), \(F_v(3,3,3;5) \leq 24\), which leads to \(F_v(6,6,6;7) \leq 726\), and \(F_v(3,3,3;8) \leq 727\).
As usual, \(K_{m,n}\) denotes the complete bipartite graph with parts of sizes \(m\) and \(n\). For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_0, a_1, \ldots, a_{n-1}, b_0, b_1, \ldots, b_{n-1}\}\) and edge set \(\{a_ib_j: 0 \leq i \leq n-1, j = i,i+1, \ldots, i+k-1 \pmod{n}\}\). A spider is a tree with at most one vertex of degree more than two, called the \({center}\) of the spider. A leg of a spider is a path from the center to a vertex of degree one. Let \(S_l(t)\) denote a spider of \(l\) legs, each of length \(t\). An \(H\)-decomposition of a graph \(G\) is an edge-disjoint decomposition of \(G\) into copies of \(H\). In this paper, we investigate the problems of \(S_l(2)\)-decompositions of complete bipartite graphs and crowns, and prove that: (1) \(K_{n,tl}\) has an \(S_l(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\), \(n \geq 2l\) if \(t = 1\), and \(n \geq 1\) if \(t \geq 2\), (2) for \(t \geq 2\) and \(n \geq tl\), \(C_{n,tl}\) has an \(S_l(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\), and (3) for \(n \geq 3t\), \(C_{n,tl}\) has an \(S_3(2)\)-decomposition if and only if \(nt \equiv 0 \pmod{2}\) and \(n \equiv 0 \pmod{4}\) if \(t = 1\).
In this paper, we extend the study on packing complete graphs \(K_v\) with \(6\)-cycles. Mainly, we obtain the maximum packing of \(K_v – L\) and a leave, where \(L\) is a vertex-disjoint union of cycles in \(K_v\).
For a vertex \(v\) of a graph \(G\), the unlabeled subgraph \(G-v\) is called a \({card}\) of \(G\). We prove that the connectedness of an \(n\)-vertex graph \(G\) and the presence of isolated vertices in \(G\) can be determined from any collection of \(n-2\) of its cards. It is also proved that if two graphs on \(n \geq 6\) vertices with minimum degree at least two have \(n-2\) cards in common, then the numbers of edges in them differ by at most one.
Let \(G\) be a connected cubic graph embedded on a surface \(\Sigma\) such that every face is bounded by a cycle of length \(6\). By Euler formula, \(\Sigma\) is either the torus or the Klein bottle. The corresponding graphs are called toroidal polyhex graphs and Klein-bottle polyhex graphs, respectively. It was proved that every toroidal polyhex graph is hamiltonian. In this paper, we prove that every Klein-bottle polyhex graph is hamiltonian. Furthermore, lower bounds for the number of Hamilton cycles in Klein-bottle polyhex graphs are obtained.
The matching preclusion number of a graph \(G\), denoted by \(mp(G)\), is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. Besides its theoretical linkage with conditional connectivity and extremal graph theory, the matching preclusion number serves as a measure of robustness in interconnection networks. In this paper, we develop general properties related to matchings in the Cartesian product of graphs, enabling us to establish the matching preclusion number for various interconnection (product) networks, specifically: hyper Petersen, folded Petersen, folded Petersen cube, hyperstar, star-cube, and hypercube. Furthermore, we show that the Cartesian product of graphs operation inherits the matching preclusion number optimality from factor graphs of even order, reinforcing the Cartesian product as a desirable network-synthesizing operator.
This paper proves that the graphic matroids with at least two edges and no isolated vertices coincide with the class of complete \(k\)-partite graphs, where, when \(k \leq 3\), no partition class has size one. It also shows that a simple rank-\(r\) binary matroid \(M\) has every two elements in a \(4\)-circuit if \(|E(M)| \geq 2^{r-1} + 2\).
Multi-sender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we constructed one multi-sender authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of this codes are also computed.
Let \(G\) be a graph with vertex set \(V\). A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V-D\) has a neighbor in \(V-D\). The minimum cardinality of a total restrained dominating set of \(G\) is called the total restrained domination number of \(G\), denoted by \(\gamma_{tr}(G)\). Cyman and Raczek \((2006)\) showed that if \(G\) is a connected graph of order \(n\) and minimum degree \(\delta\) such that \(2 \leq \delta \leq n-2\), then \(\gamma_{tr}(G) \leq n-\delta\). In this paper, we first introduce the concept of max-min total restrained domination number, denoted by \(\gamma_{tr}^M(G)\), of \(G\), and extend the above result by showing that \(\gamma_{tr}^M(G) \leq \gamma_{tr}(G) \leq n-\delta\). We then proceed to establish that \((1)\) \(\gamma_{tr}^M(G) \leq n-2\delta\) if \(n \geq 11\) and \(G\) contains a cut-vertex, and \((2)\) \(\gamma_{tr}(G) \leq n-4\) if \(n \geq 11\) and \(\delta \geq 2\).
In response surface analysis, it is generally assumed that the observations are independent and there is no effect of neighbouring units. But under the situation when the units are placed linearly with no gaps, the experimental units may experience neighbour or overlap effects from neighbouring units. Hence, for proper specification it is important to include the neighbour effects in the model. First order response surface mode! with neighbour effects from immediate left and right neighbouring units has been considered here and the conditions have been derived for the orthogonal estimation of coefficients of this model. The variance of estimated response has also been obtained and conditions for first order response surface model with neighbour effects to be rotatable have been obtained. A method of obtaining designs satisfying the derived conditions has been proposed. A first order rotatable design with neighbour effects using half replicate of \(2^3\) has also been given.
In [J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A \(118(2011) 2056-2058]\), Guo and Wang proposed a new model for disjunct matrices. As a generalization of Guo-Wang’s designs, we obtain a
new family of pooling designs. Our designs and Guo-Wang’s designs have the same numbers of items and pools, but the error-tolerance property of our design is better than that of Guo-Wang’s designs under some conditions.
A \({vertex \;irregular\; total \;labeling}\) \(\sigma\) of a graph \(G\) is a labeling of vertices and edges of \(G\) with labels from the set \(\{1, 2, \ldots, k\}\) in such a way that for any two different vertices \(x\) and \(y\), their weights \(wt(x)\) and \(wt(y)\) are distinct. The \({weight}\) \(wt(x)\) of a vertex \(x\) in \(G\) is the sum of its label and the labels of all edges incident with \(x\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total labeling is called the \({total \;vertex\; irregularity \;strength}\) of \(G\). In this paper, we study the total vertex irregularity strength for two families of graphs, namely Jahangir graphs and circulant graphs.
The Sum-Balaban index is defined as
\[SJ(G) = \frac{|E(G)|}{\mu+1} \sum\limits_{uv \in E(G)} \frac{1}{\sqrt{D_G(u)+D_G(v)}}\],
where \(\mu\) is the cyclomatic number of \(G\) and \(D_G(u)=\sum_{u\in V(G)}d_G(u,v)\). In this paper, we characterize the tree with the maximum Sum-Balaban index among all trees with \(n\) vertices and diameter \(d\). We also provide a new proof of the result that the star \(S_n\) is the graph which has the maximum Sum-Balaban index among all trees with \(n\) vertices. Furthermore, we propose a problem for further research.
A connected graph \(G = (V, E)\) is called a quasi-unicycle graph if there exists \(v_0 \in V\) such that \(G – v_0\) is a unicycle graph. Denote by \(\mathcal{G}(n, d_0)\) the set of quasi-unicycle graphs of order \(n\) with the vertex \(v_0\) of degree \(d_0\) such that \(G – v_0\) is a unicycle graph. In this paper, we determine the maximum spectral radii of quasi-unicycle graphs in \(\mathcal{G}(n, d_0)\).
Let \(Diag(G)\) and \(D(G)\) be the degree-diagonal matrix and distance matrix of \(G\), respectively. Define the multiplier \(Diag(G)D(G)\) as the degree distance matrix of \(G\). The degree distance of \(G\) is defined as \(D'(G) = \sum_{x \in V(G)} d_G(x) D(x)\), where \(d_G(u)\) is the degree of vertex \(x\), \(D_G(x)=\sum_{u\in V(G)}d_G(u,x)\) and \(d_G(u,x)\) is the distance between \(u\) and \(v\). Obviously, \(D'(G)\) is also the sum of elements of the degree distance matrix \(Diag(G)D(G)\) of \(G\). A connected graph \(G\) is a cactus if any two of its cycles have at most one common vertex. Let \(\mathcal{G}(n,r)\) be the set of cacti of order \(n\) and with \(r\) cycles. In this paper, we give the sharp lower bound of the degree distance of cacti among \(\mathcal{G}(n,r)\), and characterize the corresponding extremal cactus.
We introduce the concept of molds, which together with an appropriate weight function, gives all the information of a regular tournament. We use the molds to give a shorter proof of the characterization of domination graphs than the one given in \([4, 5]\), We also use the molds to give a lower and an upper bound of the dichromatic number for all regular tournaments with the same mold.
In this paper, we prove that every countable set of formulas of the propositional logic has at least one equivalent independent subset. We illustrate the situation by considering axioms for Boolean algebras; the proof of independence we give uses model forming.
In this paper, we introduce a new type of graph labeling known as \({super\; mean \;labeling}\). We investigate the super mean labeling for the Complete graph \(K_n\), the Star \(K_{1,n}\), the Cycle \(C_{2n+1}\), and the graph \(G_1 \cup G_2\), where \(G_1\) and \(G_2\) are super mean graphs, as well as some standard graphs.
The \({corona}\) of two graphs \(G\) and \(H\), written as \(G \odot H\), is defined as the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and joining by an edge the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae of the (modified) Schultz and Zagreb indices in the corona of two graphs.
A geodetic (resp. monophonic) dominating set in a connected graph \(G \) is any set of vertices of \(G\) which is both a geodetic (resp.monophonic) set and a dominating set in \(G\). This paper establishes some relationships between geodetic domination and monophonic domination in a graph. It also investigates the geodetic domination and monophonic domination in the join, corona and composition of
connected graphs.
Let \(G\) and \(F\) be graphs. If every edge of \(G\) belongs to a subgraph of \(G\) isomorphic to \(F\), and there exists a bijection \(\lambda: V(G) \bigcup E(G) \rightarrow \{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that the set \(\{\sum_{v\in V(F’)}\lambda(v)+\sum_{e\in E(f’)}\lambda(e):F’\cong F,F’\subseteq G\}\) forms an arithmetic progression starting from \(a\) and having common difference \(d\), then we say that \(G\) is \((a,d)\)-\(F\)-antimagic. If, in addition, \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then \(G\) is \emph{super} \((a,d)\)-\(F\)-antimagic. In this paper, we prove that the grid (i.e., the Cartesian product of two nontrivial paths) is super \((a,1)\)-\(C_4\)-antimagic.
Here presented is a unified expression of Stirling numbers and their generalizations by using generalized factorial functions and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova, Todorov, and Ahuja-Enneking are included as particular
cases of our generalization. Four algorithms for calculating the Stirling numbers and their generalizations based on our unified form are also given, which include two comprehensive algorithms using the characterization of Riordan arrays.
We give necessary and sufficient conditions to decompose \( \lambda \) copies, where necessarily \( \lambda \geq 2 \), of the complete graph \( K_v \), into so-called “2-petal”, “stem-infinity”, “barbell”, and “box-edge” graphs, all with four vertices and five edges.
The total chromatic number conjecture, which has appeared in a few hundred articles and in numerous books thus far, is now one of the classic mathematical unsolved problems. It appears that many authors coincidentally have attributed it to Professor M. Behzad and/or to Professor V.G. Vizing. Eventually, after four decades, Professor A. Soifer investigated the origin of this conjecture; published his findings in *The Mathematical Coloring Book* (2009); and stated that, “In my opinion this unquestionably merits the joint credit to Vizing and Behzad.” After checking all the arguments presented and the blames cited, I decided to investigate the controversy stated in this book on my own. My findings, which are presented in this report, specifically signify the following two points:
We will study the random perturbation on a linear differential equation as a nowhere differentiable function. The noise in the historical Langevin stochastic differential equation will be treated as a nowhere differentiable model for Brownian motion. A short introduction of Wiener process leading to It\^o’s calculus will be used in derivation of the mean and variance of the solutions to the Langevin Equation. Computational algorithms were developed and applied to study the numerical solutions to linear stochastic differential equations. Symbolic computation and simulation of a computer algebra system will be used to demonstrate the behavior of the solution to the Langevin Stochastic Differential Equation when the perturbation is density independent.
A bi-level balanced array (B-array) \( T \) with parameters \( (m, N, t) \) and index set \( \underline{\mu}’ = \{\mu_0, \mu_1, \ldots, \mu_t\} \) is a matrix with \( m \) rows, \( N \) columns, and with two elements (say, \( 0 \) and \( 1 \)) such that in every \( (t \times N) \)-submatrix \( T^* \) (clearly, there are \( \binom{m}{t} \) such submatrices) of \( T \), the following combinatorial condition is satisfied: every \( (t \times 1) \) vector \( \underline{\alpha} \) of \( T^* \) with \( i \) (\( 0 \leq i \leq t \)) ones in it appears the same number \( \mu_i \) (say) times. \( T \) is called a B-array of strength \( t \). Clearly, an orthogonal array (O-array) is a special case of a B-array. These combinatorial arrays have been extensively used in information theory, coding theory, and design of experiments. In this paper, we restrict ourselves to arrays with \( t = 4 \) and \( t = 6 \). We derive some inequalities involving \( m \) and \( \mu_i \), using the concept of coincidences amongst the columns of \( T \), which are necessary conditions for B-arrays to exist. We then use these inequalities to study the existence of these arrays and to obtain the bounds on the number of rows (also called constraints) \( m \), for a given value of \( \underline{\mu}’ \).
The typical real-time wireless video-audio digital transmission process consists of capturing the signal, digitizing it, compressing it, adding cryptography to it (crypto it), adding redundancy to enable the receiver to detect and correct a number of bit errors, packetizing it, and then transmitting it. Transmitting the signal via the Transmission Control Protocol (TCP-IP) provides a fixed number of redundancy bits, and a very rigid transmission process that could result in a large number of automatic repeat requests and denial of services. In this research, we develop a dynamic transmission algorithm, whereby the degree of redundancy is a function of the noise and the probability \( p \) for a bit to be corrupted. We also provide a variable number of protection depending on the importance of certain bits. In addition, we provide a variable packet size depending on the noise, in order to decrease the probability of automatic repeat request. The preferred protocol to be used with our algorithm is the User Datagram Protocol (UDP) fortified with our dynamic redundancy check algorithm, a packet sequence number, number of redundancy bits, signal group size as part of the packet header. Our algorithm has two parts. The first one is noise detection and noise quantization. The second part is redundancy bit adjustment and packet size adjustment to maximize the transmission throughput. In this paper, we present the analytics of keeping the correctable groups of bits in each transmission until the whole packet is received.
Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD( \( v, k, \lambda, k_1, \lambda_1 \) ), were introduced by Chan and Sarvate along with some existence results for block size \( 3 \) and \( 4 \). We have shown that necessary conditions are sufficient for the existence of BOBIBDs with \( k = 5 \) for \( k_1 = 2 \) and \( 3 \) along with partial results for \( k_1 = 4 \). We also claim the nonexistence of cyclic solutions for certain BOBIBDs. The existence of the previously unknown BOBIBD(\( v, 4, 2, 3, 1 \)), \( v \equiv 1 \pmod{6} \), is demonstrated for all \( v \geq 19 \).
Delaunay graphs have been used in CAD/CAM, sensor networks, and geographic information systems. We investigate the reliability properties of nodes in Delaunay graphs. For measuring the reliability, we formulate the concept of roaming-region for nodes. The \({roaming-region}\) \( R(i) \) of a Delaunay node \( v_i \) is such that the Delaunay graph does not change as long as \( v_i \) remains within \( R(i) \). A node \( v_i \) with a large roaming region \( R(i) \) such that \( v_i \) is positioned near the center of \( R(i) \) is identified as a reliable node. Two types of roaming regions called (i) \({lateral\; roaming\; region}\) \( LR(i) \) and (ii) \({radial\; roaming\; region}\) \( RR(i) \) are distinguished to develop the algorithm. The roaming region itself is expressed as the intersection of \( RR(i) \) and \( LR(i) \). For nodes inside the convex hull, called \({deep\; internal\; nodes}\), we present an \( O(n^2) \) time algorithm for computing their roaming region, where \( n \) is the number of nodes in the Delaunay triangulation. We finally discuss generalization and extension of the proposed algorithm.
A pencyclic graph on \( v \) vertices is called pancyclic if it contains cycles of every length from \( 3 \) to \( v \). In this paper we address the question: what is the minimum number of edges in a pancyclic graph? We present a simple analysis using chord patterns.
Software interaction test suites serve two complementary roles. They are employed to systematically verify that, for some strength \( t \), no \( t \)-way interaction of a system’s parameters causes a fault. They are also employed to locate a faulty configuration when at least one interaction fault remains. Algorithms to find such test suites employing a number of tests close to the minimum have been extensively explored, in order to test all \( t \)-way interactions. However, when faults remain, the expected number of tests needed to reveal an interaction fault is also important. One might anticipate that the test suites of minimum size also have the lowest expected time to detection of an interaction fault; or, at the very least, that some test suite of minimum size does. However, in this paper it is shown that minimum test suite size and lowest expected time to fault detection are incompatible objectives. This underlies a challenging problem of how to generate test suites that have early coverage of \( t \)-way interactions, in order to reduce time to fault detection. A hybrid approach is developed that combines a simple greedy algorithm with heuristic search to construct one test at a time while attempting to maximize the number of \( t \)-way interactions covered by the earliest tests.
Faults in software systems often occur due to interactions between parameters. Several studies show that faults are caused by 2-way through 6-way interactions of parameters. In the context of test suite prioritization, we have studied prioritization by 2-way inter-window interaction coverage and found that this criterion is effective at finding faults quickly in the test execution cycle. However, since faults may be caused by interactions between more than 2 parameters, in this paper, we provide a greedy algorithm for test suite prioritization by \( n \)-way combinatorial coverage of inter-window interactions. While greedy algorithms that generate Combinatorial Interaction Test suites enumerate and track the coverage of all possible \( t \)-tuples and constraints, we have noticed that our user-session-based test suites often do not contain every possible \( t \)-tuple, and we can take advantage of this in our algorithm by only storing \( t \)-tuples that appear in the test suite. Our empirical study shows both time and memory usage associated with our algorithm for 3-way inter-window parameter-value interaction coverage. Further, we conduct an empirical study where we compare 2-way and 3-way combinatorial coverage of inter-window parameter interactions in terms of the rate of fault detection for a web application called Schoolmate and a user-session-based test suite. Our results show that the rate of fault detection for 2-way and 3-way prioritization are within \(1\%\) of each other, but 2-way provides a slightly better result. A closer look at the characteristics of the web application, test cases, and faults reveals that most faults are triggered by 2-way interactions. We motivate the need for future work to examine a larger set of empirical studies to identify characteristics of web applications that benefit from prioritization with higher strength inter-window event interaction coverage.
In this work, we present a greedy algorithm for covering the set of incomplete STRIPS planning domain interpretations by \( t \)-strength diagnoses. We present a greedy algorithm to cover the incomplete domain model interpretations with a set of plans by iteratively generating plans so that each additional plan is biased to cover at least one new interpretation not previously covered. We also present a second greedy algorithm to construct a set of plans that covers all \( t \)-strength diagnoses of plan failure for plans in the incomplete domain model. We show that covering domain interpretations by \( t \)-strength diagnoses leads to increased coverage by a set of plans despite potentially lower coverage per plan because covering by \( t \)-strength diagnoses leads to a more scalable approach to planning where more plans can be found.
The article presents the compatibility matrix method and illustrates it with the application to the \( \text{P} \) vs \( \text{NP} \) problem. The method is a generalization of descriptive geometry: in the method, we draft problems and solve them utilizing the image creation technique. The method reveals: \( \text{P} = \text{NP} = \text{PSPACE} \subseteq \text{P/poly} \), etc.
Our previous paper [9] applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs [5] to the space of random matchings in \( K_{2n} \), deriving new proofs to a number of results on the enumeration of regular graphs with excluded cycles through the configuration model [3]. Here we extend this from excluded cycles to some excluded balanced subgraphs, and derive asymptotic results on the probability that a random regular multigraph from the configuration model contains at least one from a family of balanced subgraphs in question.
Restricted edge connectivity is a more refined network reliability index than edge connectivity. It is known that communication networks with larger restricted edge connectivity are more locally reliable.
This work presents a distance condition for graphs to be maximally restricted edge connected, which generalizes Plesník’s corresponding result.
Murty characterized the connected binary matroids with all circuits having the same size. Here we characterize the connected
bicircular matroids with all circuits having the same size.
An \(L(2,1)\)-labeling of a graph \(G\) is an assignment of nonnegative
integers to the vertices of \(G\) such that adjacent vertices get numbers
at least two apart, and vertices at distance two get distinct numbers.
The \(L(2,1)\)-labeling number of \(G\), \(\lambda(G)\), is the minimum range of
labels over all such labelings. In this paper, we determine the \(\lambda\)-
numbers of flower snark and its related graphs for all \(n \geq 3\).
In this paper, some limit relations between multivariable
Hermite polynomials \((MHP)\) and some other multivariable polyno-
mials are given, a class of multivariable polynomials is defined via
generating function, which include \((MHP)\) and multivariable Gegen-
bauer polynomials \((MGP)\) and with the help of this generating func-
tion various recurrence relations are obtained to this class. Integral
representations of \(MHP\) and \(MGP\) are also given. Furthermore, gene-
ral families of multilinear and multilateral generating functions are
obtained and their applications are presented.
We give some properties of skew spectrum of a graph, especially,
we answer negatively a problem concerning the skew characteristic
polynomial and matching polynomial in [M. Cavers et al., Skew-
adjacency matrices of graphs, Linear Algebra Appl. \(436 (2012) 4512-
4529]\).
This paper is devoted to studying the form of the solutions and
the periodicity of the following rational system of difference
equations:
\begin{align*}
x_{n+1} &= \frac{x_{n-5}}{1-x_n-_5y_{n-2}}, &
y_{n+1}= \frac{ y_{n-5}}{\pm1 \pm y_{n-5} + _5x_{n-2}},
\end{align*}
with initial conditions are real numbers.
The Moore bound states that a digraph with maximum out-degree \(d\)
and radius \(k\) has at most \(1 + d + \cdots + d^k\) vertices.
Regular digraphs attaining this bound and whose diameter is at most
\(k + 1\) are called radially Moore digraphs. Körner [4] proved
that these extremal digraphs exist for any value of \(d \geq 1\) and \(k \geq 1\).
In this paper, we introduce a digraph operator based on the line
digraph, which allows us to construct new radially Moore digraphs
and recover the known ones. Furthermore, we show that for \(k = 2\),
a radially Moore digraph with as many central vertices as the degree
\(d\) does exist.
The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\)
is the set consisting of \(e\) and of all edges having a common
end-vertex with \(e\) . Let \(f\) be a function on \(E(G)\) , the edge
set of \(G\) , into the set \(\{-1, 0, 1\}\). If \(\sum_{x \in N_G[e]} f(x) \geq 1\)
for each \(e \in E(G)\), then \(f\) is called a minus edge
dominating function of \(G\).
The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over
all minus edge dominating functions \(f\) of \(G\), is called the
\emph{minus edge domination number} of \(G\) and is denoted by
\(\gamma’_m(G)\).
It has been conjectured that \(\gamma’_m(G) \geq n – m\) for every
graph \(G\) of order \(n\) and size \(m\). In this paper, we prove
that this conjecture is true and then classify all graphs \(G\)
with \(\gamma’_m(G) = n – m\).
We seek a decomposition of a complete equipartite graph minus
a one-factor into parallel classes each consisting of cycles of length
\(k\). In this paper, we address the problem of resolvably decomposing
complete multipartite graphs with \(r\) parts each of size \(\alpha\) with a one-
factor removed into \(k\)-cycles. We find the necessary conditions, and
give solutions for even cycle lengths.
An adjacent vertex distinguishing edge coloring, or an avd-coloring,
of a simple graph \(G\) is a proper edge coloring of \(G\) such that
no two adjacent vertices are incident with the same set of colors.
H. Hatami showed that every simple graph \(G\) with no isolated
edges and maximum degree \(\Delta\) has an avd-coloring with at
most \(\Delta + 300\) colors, provided that \(\Delta > 10^{20}\).
We improve this bound as follows: if \(\Delta > 10^{15}\), then the
avd-chromatic number of \(G\) is at most \(\Delta + 180\), where
\(\Delta\) is the maximum degree of \(G\).
The Padmakar-Ivan (\(PI\)) index of a graph \(G = (V, E)\) is defined
as \(PI(G) = \sum_{e \in uv} (n_{eu}(e|G) + n_{ev}(e|G))\)
where \(n_{eu}(e|G)\) is the number of edges of \(G\) lying closer to \(u\)
than to \(v\) and \(n_{ev}(e|G)\) is the number of edges of \(G\) lying
closer to \(v\) than to \(u\).
In this paper, we derive a recursive formula for computing the
\(PI\) index of a double hexagonal chain using the orthogonal cut,
and characterize the double hexagonal chains with extremal
\(PI\) indices.
In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is
performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We define the
pegging number as the smallest number of pegs needed to reach all the vertices in a graph no matter what the distribution. Similarly, the optimal-pegging number of a graph is defined as the smallest distribution of pegs for which all the vertices in the graph can be reached.We obtain tight bounds on the pegging numbers and optimal-pegging numbers of complete binary trees and compute the optimal-pegging numbers of complete infinitary trees. As a result of these computaions, we deduce that there is a tree whose optimal-pegging number is strictly increased by removing a leaf. We also compute the optimal-pegging number of caterpillar graphs and the tightest upper boundon the optimal-pegging numbers of lobster graphs.
The Laplacian-energy-like graph invariant of a graph \(G\), denoted by \(LEL(G)\), is defined as \(LEL(G) = \sum\limits_{i=1}^{n} \sqrt{\mu_i}\), where \(\mu_i\) are the Laplacian eigenvalues of graph \(G\). In this paper, we study the maximum \(LEL\) among graphs with a given number of vertices and matching number. Some results on \(LEL(G)\) and \(LEL(\overline{G})\) are obtained.
In this paper, we consider mixed arrangements, which are composed of
hyperplanes (or subspaces) and spheres. We investigate the posets of
their intersection sets and calculate the Möbius functions of the
mixed arrangements through the hyperplane (or subspace) arrangements’
Möbius functions. Furthermore, by employing the method of deletion
and restriction, we derive recursive formulas for the triples of
these mixed arrangements.
For every integer \(c\), let \(n = R_d(c)\) be the least integer such
that for every coloring \(\Delta: \{1, 2, \ldots, 2n\} \to \{0, 1\}\),
there exists a solution \((x_1, x_2, x_3)\) to
\[x_1 + x_2 + x_3 = c\]
such that \(x_i \neq x_j\) when \(i \neq j\),
and
\(\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\).
In this paper, it is shown that for every integer \(c\),
\[R_d(c) =
\begin{cases}
4c + 8 & \text{if } c \geq 1,\\
8 & \text{if } -3 \leq c < -6,\\
9 & \text{if} c=0,-2,-7,-8\\
10 & \text{if } c =-1,-9 \\
|c| -\left\lfloor \frac{|c|-4}{5} \right\rceil & \text{if } c \leq -10.
\end{cases}\]
A graph \(G\) with an even number of vertices is said to be
almost self-complementary if it is isomorphic to one of its
almost complements \(G^c – M\), where \(M\) denotes a perfect matching
in its complement \(G^c\). In this paper, we show that the diameter
of connected almost self-complementary graphs must be \(2\), \(3\), or
\(4\). Furthermore, we construct connected almost self-complementary
graphs with \(2n\) vertices having diameter \(3\) and \(4\) for each \(n \geq 3\),
and diameter \(2\) for each \(n \geq 4\), respectively. Additionally, we
also obtain that for any almost self-complementary graph \(G_n\) with
\(2n\) vertices, \(\lceil \sqrt{n}\rceil \leq \chi(G_n) \leq n\). By
construction, we verify that the upper bound is attainable for each
positive integer \(n\), as well as the lower bound when \(\sqrt{n}\)
is an integer.
A \(k\)-container \(C(u,v)\) in a graph \(G\) is a set of \(k\) internally
vertex-disjoint paths between vertices \(u\) and \(v\). A \(k^*\)-container
\(C(u,v)\) of \(G\) is a \(k\)-container such that \(C(u,v)\) contains all
vertices of \(G\). A graph is globally \(k^*\)-connected if there exists
a \(k^*\)-container \(C(u,v)\) between any two distinct vertices \(u\) and \(v\).
A \(k\)-regular graph \(G\) is super \(k\)-spanning connected if \(G\) is
\(i^*\)-connected for \(1 \leq i \leq k\). A graph \(G\) is \(1\)-fault-tolerant
Hamiltonian if \(G – F\) is Hamiltonian for any \(F \subseteq V(G)\) and
\(|F| = 1\). In this paper, we prove that for cubic graphs, every
super \(3\)-spanning connected graph is globally \(3^*\)-connected and
every globally \(3^*\)-connected graph is \(1\)-fault-tolerant Hamiltonian.
We present examples of super \(3\)-spanning connected graphs, globally
\(3^*\)-connected graphs that are not super \(3\)-spanning connected,
\(1\)-fault-tolerant Hamiltonian graphs that are globally \(1^*\)-connected
but not globally \(3^*\)-connected, and \(1\)-fault-tolerant Hamiltonian
graphs that are neither globally \(1^*\)-connected nor globally \(3^*\)-connected.
Furthermore, we prove that there are infinitely many graphs in each
such family.
In this paper, we prove a fixed point theorem for weakly compatible mappings satisfying a general contractive condition of operator type. In short, we are going to study mappings \( A, B, S, T: X \to X \) for which there exists a right continuous function \( \psi: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \(\psi(0) = 0\) and \(\psi(s)\leq s\) for \(s > 0.\) Moreover, for each \( x, y \in X \), one has \(O(f; d(Sx, Ty)) \leq \psi(O(f; M(x,y))),\) where \( O(f; \cdot) \) and \( f \) are defined in the first section. Also in the first section, we give some examples for \( O(f; \cdot) \). The second section contains the main result. In the last section, we give some corollaries and remarks.
We consider unitary graphs attached to \(\mathbb{Z}^{d}_{n}\) using an analogue of the Euclidean distance. These graphs are shown to be integral when \(d\) is odd or the dimension \(d\) is even.
A graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we determine the graph with the
largest spectral radius among all connected cactuses with n vertices and edge independence number \(q\).
In this study, we obtained lower and upper bounds for the Euclidean norm of a complex matrix \(A\) of order \(n \times n\). In addition,
we found lower and upper bounds for the spectral norms and Euclidean norms of the Hilbert matrix its Hadamard
square root, Cauchy-Toeplitz and Cauchy-Hankel matrices in the forms \(H = \left(\frac{1}{i + j – 1}\right)_{i,j=1}^n\),\(H^{\frac{01}{2}}=(\frac{1}{(i+j-1)}^{\frac{1}{2}})_{i,j=1}^n\); \(T_n = \left[\frac{1}{(g+(i + j)h)}_{i,j=1}^n\right]\), and \(H_n = \left[\frac{1}{(g+(i + j )h}\right]_{i,j=1}^n\), respectively.
Let \(G\) be a graph, and let \(a\) and \(b\) be nonnegative integers such that \(1 \leq a \leq b\). Let \(g\) and \(f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \((g, f)\)-factor if \(g(x) \leq d_G^h(x) \leq f(x)\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy \in E(G)\}\). The isolated toughness \(I(G)\) of a graph \(G\) is defined as follows: If \(G\) is a complete graph, then \(I(G) = +\infty\); else, \(I(G) = \min\{ \frac{|S|}{i(G-S)} : S \subseteq V(G), i(G – S) \geq 2 \}\), where \(i(G – S)\) denotes the number of isolated vertices in \(G – S\). In this paper, we prove that \(G\) has a fractional \((g, f)\)-factor if \(\delta(G) \geq I(G) \geq \frac{b(b-1)}{a}+1\). This result is best possible in some sense.
In this paper we prove that there exists one type of connected cubic graph,which minimizes the number of spanning trees over all other connected cubic graphs of the same order \(7\), \(n\geq 14\).
Let \(T = PSL(n, q)\) be a projective linear simple group, where \(n \geq 2\),\(q\) a prime power and \((n,q) \neq (2,2)\) and \((2,3)\). We classify all \(3— (v, k, 1)\) designs admitting an automorphism group \(G\) with \(T \unlhd G \leq Aut(T)\) and \(v=\frac{q^n-1}{q-1}.\)
In this paper, we introduce the notion of \(f\)-derivations and investigate the properties of \(f\)-derivations of lattice implication
algebras. We provide an equivalent condition for an isotone \(f\)-derivation in a lattice implication algebra. Additionally, we
characterize the fixed set \({Fix_d}(L)\) and \(\mathrm{Kerd}\) by \(f\)-derivations. Furthermore, we introduce
normal filters and obtain some properties of normal filters in lattice implication algebras.
We give a new combinatorial interpretation of Lah and \(r\)-Lah numbers.
We establish two cross recurrence relations: the first one, which uses
an algebraic approach, is a recurrence relation of order two with
rational coefficients; the second one uses a combinatorial proof and
is a recurrence relation with integer coefficients. We also express
\(r\)-Lah numbers in terms of Lah numbers. Finally, we give identities
related to rising and falling factorial powers.
In this paper, we reveal the yin-yang structure of the affine plane of order four by characterizing the unique blocking set as the
Mébius-Kantor configuration \(8_3\).
A family of sets is called \(K\)-union distinct if all unions involving \(K\) or fewer members thereof are distinct. If a family of
sets is \(K\)-cover-free, then it is \(K\)-union distinct. In this paper, we recognize that this is only a sufficient condition and,
from this perspective, consider partially cover-free families of sets with a view to constructing union distinct families. The
role of orthogonal arrays and related combinatorial structures is explored in this context. The results are applied to find
efficient anti-collusion digital fingerprinting codes.
Let \(G\) be a \(2\)-edge-connected simple graph on \(n\) vertices, \(n \geq 3\). It is known that if \(G\) satisfies \(d(x) \geq \frac{n}{2}\) for every vertex \(x \in V(G)\), then \(G\) has a nowhere-zero \(3\)-flow, with several exceptions.In this paper, we prove that, with ten exceptions, all graphs with at most two vertices of degree less than \(\frac{n}{2}\) have nowhere-zero \(3\)-flows. More precisely, if \(G\) is a \(2\)-edge-connected graph on \(n\) vertices, \(n \geq 3\), in which at most two vertices have degree less than \(\frac{n}{2}\), then \(G\)
has a nowhere-zero \(3\)-flow if and only if \(G\) is not one of ten completely described graphs.
In this paper, we introduce the notion of right derivation of a weak BCC-algebra and investigate its related properties.
Additionally, we explore regular right derivations and d-invariants on weak BCC-ideals in weak BCC-algebras.
We investigate the Jacobsthal numbers \(\{J_n\}\) and Jacobsthal-Lucas numbers \(\{j_n\}\). Let \(\mathcal{J}_n = J_n \times j_n\) and \(\mathcal{J}_n = J_n + j_n\).In this paper, we give some determinantal and permanental representations for \(\mathcal{J}_n\) and \(\mathcal{J}_n\). Also, complex factorization formulas for the numbers are presented.
Let \(d\) be a fixed integer, \(0 \leq d \leq 2\), and let \(\mathcal{K}\) be a family of sets in the plane having simply connected union. Assume that for every countable subfamily \(\{K_n : n \geq 1\}\) of \(\mathcal{K}\), the union \(\cup\{K_n \geq 1\}\) is
starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K:K \in \mathcal{K}\}\) has these properties as well.
In the finite case ,define function \(g\) on \((0, 1, 2) \) by \(g(0) = 2\), \(g(1) = g(2) = 4\). Let \(\mathcal{K}\) be a finite family of nonempty compact sets in the plane such that \(\cup\{K \in \mathcal{K}\}\) has a connected complement. For fixed \(d \in \{0, 1, 2\}\), assume that for every \(g(d)\) members of \(\mathcal{K}\), the corresponding union is starshaped via staircase paths and its staircase kernel contains a convex set of dimension at least \(d\). Then, \(\cup\{K \in \mathcal{K}\}\) also has these properties,also.
Most of these results are dual versions of theorems that hold for intersections of sets starshaped via staircase paths.The exceotion is the finite case above when \(d = 2\) .Surprisingly ,although the result for \(d=2\) holds for unique of sets, no analogue for intersections of sets is possible.
Let \(G\) be a simple connected graph containing a perfect matching.
\(G\) is said to be BM-extendable (bipartite matching extendable)
if every matching \(M\) which is a perfect matching of an induced
bipartite subgraph of \(G\) extends to a perfect matching of \(G\).
The BM-extendable cubic graphs are known to be \(K_{4}\) and \(K_{3,3}\).
In this paper, we characterize the 4-regular BM-extendable graphs.
We show that the only 4-regular BM-extendable graphs are \(K_{4,4}\) and
\(T_{4n}\), \(n \geq 2\), where \(T_{4n}\) is the graph on \(4n\) vertices
\(u_{i}\), \(v_{i}\), \(x_{i}\), \(y_{i}\), \(1 \leq i \leq n\), such that
\(\{u_{i}, v_{i}, x_{i}, y_{i}\}\) is a clique and
\(x_{i}u_{i+1}\), \(y_{i}v_{i+1} \in E(T_{4n})\) (mod \(n\)).
A rainbow coloring of the edges of a graph is a coloring such
that no two edges of the graph have the same color. The
anti-Ramsey number \(f(G, H)\) is the maximum number of colors
such that there is an \(H\)-anti-Ramsey edge coloring of \(G\), that is,
there exists no rainbow copy of the subgraph \(H\) of \(G\) in some
coloring of the edges of the host graph \(G\) with \(f(G, H)\) colors.
In this note, we exactly determine \(f(Q_5, Q_2)\) and \(f(Q_5, Q_3)\),
where \(Q_n\) is the \(n\)-dimensional hypercube.
The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum
of weights \(\frac{2}{d(u) + d(v)}\) of all edges \(uv\) of \(G\), where
\(d(u)\) denotes the degree of a vertex \(u\) in \(G\).
In this paper, we establish sharp lower and upper bounds for the
harmonic index of bicyclic graphs and characterize the
corresponding extremal graphs.
For a graph \(G\), its Hosoya index is defined as the total number
of matchings in it, including the empty set. As one of the oldest and
well-studied molecular topological descriptors, the Hosoya index has
been extensively explored.
Notably, existing literature has primarily focused on its extremal
properties. In this note, we bridge a significant gap by establishing
sharp lower bounds for the Hosoya index in terms of other topological
indices.
We present a unified extension of alternating subsets to \(k\)-combinations
of \(\{1, 2, \ldots, n\}\) containing a prescribed number of sequences
of elements of the same parity. This is achieved by shifting attention
from parity-alternating elements to pairs of adjacent elements of the
same parity.
Enumeration formulas for both linear and circular combinations are
obtained by direct combinatorial arguments. The results are applied
to the enumeration of bit strings.
For a graph \(G\), let \(\mathcal{D}(G)\) be the set of all strong orientations of \(G\).
Define the orientation number of \(G\), \(\overrightarrow{d}(G) = \min\{d(D) \mid D \in \mathcal{D}(G)\}\),
where \(d(D)\) denotes the diameter of the digraph \(D\).
In this paper, it is shown that \(\overrightarrow{d}(G(n_1, n_2, \ldots, n_p)) = d(G)\),
where \(G(n_1, n_2, \ldots, n_p)\) is a \(G\)-vertex multiplication
([2]) of a connected bipartite graph \(G\) of order \(p \geq 3\)
with diameter \(d(G) \geq 5\) and any finite sequence \(\{n_1, n_2, \ldots, n_p\}\)
with \(n_i \geq 3\).
Cyclic frames, or partially partition-type cyclic relative difference
families, are combinatorial structures that are used to produce series
of optimal families consisting of a single frequency hopping sequence
and optimal difference systems of sets for code synchronization.
In this paper, two new classes of cyclic frames from finite geometries
are obtained.
Consider the game of locating a marked vertex on a connected graph,
where the player repeatedly chooses a vertex of the graph as a probe,
and is given the distance from the probe to the marked vertex,
until she can uniquely locate the hidden vertex. The goal is to
minimize the number of probes.
The static version of this game is the well-known problem of finding
the metric dimension (or location number ) of the graph.
We study the sequential version of this game, and the corresponding
sequential location number .
We establish several formulae for sums and alternating sums of products
of generalized Fibonacci and Lucas numbers. In particular, we extend
some results of Z. Cerin and of Z. Cerin and G. M. Gianella .
An \({H}_2\) graph is a multigraph on three vertices with a double
edge between a pair of distinct vertices and single edges between
the other two pairs. In this paper, we settle the \({H}_2\) graph
decomposition problem, which was left unfinished in a paper of
Hurd and Sarvate, by decomposing a complete multigraph \(3K_{8t}\)
into \({H}_2\) graphs recursively.
This article is a contribution to the study of the automorphism groups
of \(2\)-\((v,k,1)\) designs. Let \(\mathcal{D}\) be a \(2\)-\((v,13,1)\) design and
suppose that \(G\) is a group of automorphisms of \(\mathcal{D}\) which is
block-transitive and point-primitive. Then \(\mathrm{Soc}(G)\),
the socle of \(G\), is not isomorphic to \(^2G_2(q)\) or to \(^2F_4(q^2)\)
for any prime power \(q\).
Let \(G\) be a finite permutation group acting primitively on sets \(\Omega_1\) and \(\Omega_2\). We describe a construction of a \(1\)-design
with the block set \(\mathcal{B}\) and the point set \(\Omega_2\), having \(G\) as an automorphism group.Applying this method, we construct a unital \(2\)-\((q^3+1, q+1, 1)\) design and a semi-symmetric design \((q^4-q^3+q^2, q^2-q, (1))\) from the unitary group \(U(3,q)\), where \(q = 3, 4, 5, 7\).From the unital and the semi-symmetric design, we build a projective plane \(PG(2,q^2)\). Further, we describe other combinatorial structures constructed from these unitary groups.
Given a (directed) graph \(G = (V,A)\), the induced subgraph of \(G\) by a subset \(X\) of \(V\) is denoted by \(G[X]\). A graph \(G = (V, A)\) is a \({tournament}\) if for any distinct vertices \(x\) and \(y\) of \(G\), \(G[\{x, y\}]\) possesses a single arc. With each graph \(G = (V,A)\), associate its \({dual}\) \(G^* = (V, A^*)\) defined as follows: for \(x,y \in V\), \((x,y) \in A^*\) if \((y,x) \in A\). Two graphs \(G\) and \(H\) are \({hemimorphic}\) if \(G\) is isomorphic to \(H\) or to \(H^*\). Moreover, let \(k > 0\). Two graphs \(G = (V,A)\) and \(H = (V,B)\) are \({k\;-hemimorphic}\) if for every \(X \subseteq V\), with \(|X| \leq k\), \(G[X]\) and \(H[X]\) are hemimorphic. A graph \(G\) is \({k\;-forced}\) when \(G\) and \(G^*\) are the only graphs \(k\)-hemimorphic to \(G\). Given a graph \(G = (V,A)\), a subset \(X\) of \(V\) is an \({interval}\) of \(G\) provided that for \(a,b \in X\) and \(x \in V\setminus X\), \((a,x) \in A\) if and only if \((b,x) \in A\), and similarly for \((x,a)\) and \((x,b)\). For example, \(\emptyset\), \(\{x\}\), where \(x \in V\), and \(V\) are intervals called trivial. A graph \(G = (V, A)\) is \({indecomposable}\) if all its intervals are trivial. Boussairi, Tle, Lopez, and Thomassé \([2]\) established the following duality result. An indecomposable graph which does not contain the graph \(({0, 1, 2}, {(0, 1), (1,0), (1,2)})\) and its dual as induced subgraphs is \(3\)-forced. A simpler proof of this theorem is provided in the case of tournaments and also in the general case. The \(3\)-forced graphs are then characterized.
Let \(G_i\) be the subgraph of \(G\) whose edges are in the \(i\)-th color in an \(r\)-coloring of the edges of \(G\). If there exists an \(r\)-coloring of the edges of \(G\) such that \(H_i \cong G_i\) for all \(1 \leq i \leq r\), then \(G\) is said to be \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). The multicolor Ramsey number \(R(H_1, H_2, \ldots, H_r)\) is the smallest integer \(n\) such that \(K_n\) is not \(r\)-colorable to \((H_1, H_2, \ldots, H_r)\). Let \(C_m\) be a cycle of length \(m\). The four-color Ramsey numbers related to \(C_6\) are studied in this paper. It is well known that \(18 \leq R_4( C_6) \leq 21\). We prove that \(R(C_5, C_4, C_4, C_4) = 19\) and \(18 \leq R(C_6, C_6, H_1, H_2) \leq 20\), where \(H_i\) are isomorphic to \(C_4\) or \(C_6\).
A graph \(G\) is called an \(M_r(k)\)-graph if \(G\) has no \(k\)-list assignment to its vertices with exactly \(r\) vertex colorings. We characterize all \(M_3(2)\)-graphs. More precisely, it is shown that a connected graph \(G\) is an \(M_3(2)\)-graph if and only if each block of \(G\) is a complete graph with at least three vertices.
A global boundary defensive \(k\)-alliance in a graph \(G = (V, E)\) is a dominating set \(S\) of vertices of \(G\) with the property that every vertex in \(S\) has \(\geq k\) more neighbors in \(S\) than it has outside of \(S\). A global boundary offensive \(k\)-alliance in a graph \(G\) is a set \(S\) of vertices of \(G\) with the property that every vertex in \(V \setminus S\) has \(k\) more neighbors in \(S\) than it has outside of \(S\). We define a global boundary powerful \(k\)-alliance as a set \(S\) of vertices of \(G\), which is both global boundary defensive \(k\)-alliance and global boundary offensive \((k+2)\)-alliance. In this paper, we study mathematical properties of boundary powerful \(k\)-alliances. In particular, we obtain several bounds (closed formulas for the case of regular graphs) on the cardinality of every global boundary powerful \(k\)-alliance. Additionally, we consider the case in which the vertex set of a graph \(G\) can be partitioned into two boundary powerful \(k\)-alliances, showing that, in such a case, \(k = -1\) and, if \(G\) is \(\delta\)-regular, its algebraic connectivity is equal to \(\delta + 1\).
We present two recursive enumeration formulas for the number of labelled essential graphs. The enumeration parameters of the first formula are the number of vertices, chain components, and cliques, while the enumeration parameters of the second formula are the number of vertices and cliques.Both formulas may be used to count the number of labelled essential graphs
with given number of vertices.
In this paper, we first survey the connections between Bell polynomials (numbers) and the derangement polynomials (numbers). Their close relations are mainly based on Hsu’ summation formula. According to this formula, we present some new identities involving harmonic numbers,Bell polynomials (numbers) and the derangement polynomials (numbers).Moreover, we find that the series \(\sum_{m\geq0}(\frac{D_m}{m!}-\frac{1}{e})\) is (absolutely) convergent and their sums are also determined, where \(D_m\) is the \(mth\) derangement number.
A graph \(G\) is regular if the degree of each vertex of \(G\) is d and almost regular or more precisely a \((d,d + 1)\)-graph, if the degree of each vertex of \(G\) is either \(d\) or \(d+1\). If \(d \geq 2\) is an integer, \(G\) a triangle-free \((d,d + 1)\)-graph of order n without an odd component and \(n \leq 4d\), then we show in this paper that \(G\) contains a perfect matching. Using a new Turdn type result, we present an analogue for triangle-free regular graphs. With respect to these results, we construct smallest connected, regular and almost regular triangle-free even order graphs without perfect matchings.
In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph \(G\) into a new graph \(\mu(G)\), which is called the Mycielskian of \(G\).This paper shows that:
For a strongly connected digraph \(D\) with \(|V(D)| \geq 2\):\(\mu(D)\) is super-\(\kappa\) if and only if \(\delta(D) < 2\kappa(D)\).;\(\mu(D)\) is super-\(\lambda\) if and only if \(D \ncong \overrightarrow{K_2}\).
The sum of the squares of eccentricity \((SSE)\) over all vertices of a connected graph is a new graph invariant proposed in \([13]\) and further studied in \([14, 15]\). In this paper, we report some further mathematical properties of \(SSE\). We give sharp lower bounds for \(SSE\) among all \(n\)-vertices connected graphs with given independence number, vertex-, and edge-connectivity, respectively. Addtionally, we give explicit formulas for \(SSE\) of Cartesian product of two graphs, from which we deduce \(SSE\) of \(C_4\), nanotube and nanotorus.
The vertex linear arboricity \(vla(G)\) of a nonempty graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths.An integer distance graph is a graph \(G(D)\) with the set of all integers as vertex set and two vertices \(u,v \in {Z}\) are adjacent if and only if \(|u-v| \in D\), where the distance set \(D\) is a subset of the positive integers.Let \(D_{m,k,3} = [1,m] \setminus \{k, 2k, 3k\}\) for \(m \geq 4k \geq 4\). In this paper, we obtain some upper and lower bounds of the vertex linear arboricity of the integer distance graph \(G(D_{m,k,3})\) and the exact value of it for some special cases.
In this paper, we generalize to the class of signed graphs the well known result that every numbered graph can be embedded as an induced subgraph in a gracefully numbered graph.
There are \(267\) nonisomorphic groups of order \(64\). It was known that \(259\) of these groups admit \((64, 28, 12)\) difference sets and the other eight groups do not admit \((64, 28, 12)\) difference sets. Despite this result, no research investigates the problem of finding all \((64, 28, 12)\) difference sets in a certain group of order \(64\).In this paper, we find all \((64, 28, 12)\) difference sets in \(111\) groups of order \(64. 106\) of these groups are nonabelian. The other five are \(\mathbb{Z}_{16} \times \mathbb{Z}_4\), \(\mathbb{Z}_{16} \times \mathbb{Z}_2^2\), \(\mathbb{Z}_8 \times \mathbb{Z}_8\), \(\mathbb{Z}_8 \times \mathbb{Z}_4 \times \mathbb{Z}_2\), and \(\mathbb{Z}_8 \times \mathbb{Z}_2^3\).In these \(111\) groups, we obtain \(74,922\) non-equivalent \((64, 28, 12)\) difference sets. These difference sets provide at least \(105\) nonisomorphic symmetric \((64, 28, 12)\) designs. Most of our work was done using programs with the software \(GAP\).
In this paper, we obtain some generating functions for the generalized Zernike or disk polynomials \(P_{m,n}^\alpha (z,z^*)\) which are investigated by Wiinsche [13]. We derive various families of bilinear and bilateral generating functions. Furthermore, some special cases of the results presented in this study are indicated. Also, it is possible to obtain multilinear and multilateral generating functions for the polynomials \(P_{m,n}^\alpha (z,z^*)\).
A \((k,t)\)-list assignment \(L\) of a graph \(G\) is a list of \(k\) colors available at each vertex \(v\) in \(G\) such that \(|\bigcup_{v\in V(G)}L(v)| = t\). A proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\) is said to be an \(L\)-coloring. We say that a graph \(G\) is \(L\)-colorable if \(G\) has an \(L\)-coloring. A graph \(G\) is \((k,t)\)-choosable if \(G\) is \(L\)-colorable for every \((k,t)\)-list assignment \(L\).
Let \(G\) be a graph with \(n\) vertices and \(G\) does not contain \(C_5\) or \(K_{k-2}\) and \(K_{k+1}\). We prove that \(G\) is \((k, kn – k^2 – 2k)\)-choosable for \(k \geq 3\).\(G\) is not \((k, kn – k^2 – 2k)\)-choosable for \(k = 2\).This result solves a conjecture posed by Chareonpanitseri, Punnim, and Uiyyasathian [W. Chareonpan-itseri, N. Punnim, C. Uiyyasathian, On \((k,t)\)-choosability of Graphs: Ars Combinatoria., \(99, (2011) 321-333]\).
We call a graph \(G\) a \({generalized \;split \; graph}\) if there exists a core \(K\) of \(G\) such that \(V(G) \setminus V(K)\) is an independent set of \(G\).Let \(G\) be a generalized split graph with a partition \(V(G) = K \cup S\), where \(K\) is a core of \(G\) and \(S\) is an independent set. We prove that \(G\) is end-regular if and only if for any \(a, b \in S\), \(\phi \in \text{Aut}(K)\), the inclusion \(\phi(N(a)) \subsetneqq N(b)\) does not hold.
\(G\) is end-orthodox if and only if \(G\) is end-regular and for any \(a, b \in S\), \(N(a) \neq N(b)\).
In this paper we generalize the Fibonacci numbers and the Lucas numbers with respect to \(n\), respectively \(n+1\) parameters. Using these definitions we count special subfamilies of the set of \(n\) integers. Next we give the graph interpretations of these numbers with respect to the number of \(P_k\),-matchings in special graphs and we apply it for proving some identity and also for counting other subfamilies of the set of n integers.
The Wiener-Hosoya index was firstly introduced by M. Randié¢ in \(2004\). For any tree \(T\), the Wiener-Hosoya index is defined as
\[WH(T)= \sum\limits_{e\in E(T)} (h(e) + h[e])\]
where \(e = uv\) is an arbitrary edge of \(T\), and \(h(e)\) is the product of the numbers of the vertices in each component of \(T – e\), and \(h[e]\) is the product of the numbers of the vertices in each component of \(T- \{u,v\}\). We shall investigate the Wiener-Hosoya index of trees with diameter not larger than \(4\), and characterize the extremal graphs in this paper.
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and
successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in
first part, and, we generalize the results obtained in \([4]\) in second part.
Fouquet and Jolivet conjectured that if \(G\) is a \(k\)-connected \(n\)-vertex graph with independence number \(\alpha \geq k \geq 2\), then \(G\) has circumference at least \( \frac{k(n+\alpha-k)}{\alpha} \). This conjecture was recently proved by \(O\), West, and Wu.
In this note, we consider the set of \(k\)-connected \(n\)-vertex graphs with independence number \(\alpha > k \geq 2\) and circumference exactly \( \frac{k(n+\alpha-k)}{\alpha} \). We show that all of these graphs have a similar structure.
Let \(\Gamma\) be the rank three \(M_{24}\) maximal \(2\)-local geometry. For the two conjugacy types of involution in \(M_{24}\), we describe the fixed point sets of chambers in \(\Gamma\).
In this paper, all connected graphs with the fourth largest signless-Laplacian eigenvalue less than two are determined.
The Lights Out game on a graph \(G\) is played as follows. Begin with a (not necessarily proper) coloring of \(V(G)\) with elements of \(\mathbb{Z}_2\). When a vertex is toggled, that vertex and all adjacent vertices change their colors from \(0\) to \(1\) or vice-versa. The game is won when all vertices have color \(0\). The winnability of this game is related to the existence of a parity dominating set.
We generalize this game to \(\mathbb{Z}_k\), \(k \geq 2\), and use this to define a generalization of parity dominating sets. We determine all paths, cycles, and complete bipartite graphs in which the game over \(\mathbb{Z}_k\) can be won regardless of the initial coloring, and we determine a constructive method for creating all caterpillar graphs in which the Lights Out game cannot always be won.
A total coloring of a simple graph \(G\) is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color.The minimum number of colors required for a proper total coloring of \(G\) is called the total chromatic number of \(G\) and denoted by \(\chi_t(G)\). The Total Coloring Conjecture (TCC) states that for every simple graph \(G\),\(\Delta(G) + 1 \leq \chi_t(G) \leq \Delta(G) + 2.\) \(G\) is called Type \(1\) (resp. Type \(2\)) if \(\chi_t(G) = \Delta(G) +1\) (resp. \(\chi_t(G) = \Delta(G) + 2\)). In this paper, we prove that the folded hypercubes \(FQ_n\), is of Type \(1\) when \(n \geq 4\).
Let \(H\) be a simple graph with \(n\) vertices and \(\mathcal{G} = \{G_1, G_2, \ldots, G_n\}\) be a sequence of \(n\) rooted graphs.
Following Godsil and McKay (Bull. Austral. Math. Soc. \(18 (1978) 21-28\)) defined the the rooted product \(H({G})\) of \(H\) by \({G}\) is defined by identifying the root of \(G_i\) with the \(i\)th vertex of \(H\).In this paper, we calculate the Wiener index of \(H({G})\), i.e., the sum of distances between all pairs of vertices, in terms of the Wiener indices of \(G_i\), \(i = 1, 2, \ldots, k\).As an application, we derive a recursive relation for computing the Wiener index of Generalized Bethe trees.
Let \(G\) be a connected graph with \(p\) vertices and \(q\) edges.A \(\gamma\)-labeling of \(G\) is a one-to-one function f from \(V(G)\) to \({0,1,…,q}\) that induces a labeling \(f’\) from \(V(G)\) to \({1,2,…,q}\) defined by \(f(e) = |f(u) – f(v)|\) for each edge \(e = uv\) of \(G\). The value of a \(\gamma\)-labeling \(f\) is defined to be the sum of the values of \(f’\) over all
edges. Also, the maximum value of a \(\gamma\)-labeling of \(G\) is defined as the maximum of the values among all \(\gamma\)-labelings of \(G,\) while the minimum value is the minimum of the values among all \(\gamma\)-labelings
of \(G\). In this paper, the maximum value and minimum value are determined for any complete bipartite graph.
A labeling f of a graph G is a bijection from its edge set \(E(G)\) to the set \(\{1, 2, …, |E(G)|\}\), which is antimagic if for any distinct vertices \(x\) and \(y\), the sum of the labels on edges incident to \(x\) is different from the sum of the labels on edges incident to \(y\). A graph G is antimagic if \(G\) has an f which is antimagic. Hartsfield and Ringel conjectured in \(1990\)
that every connected graph other than Ko is antimagic. In this paper, we show that some graphs with regular subgraphs are antimagic.
In \([1]\), the author provided a Gray code for the set of \(n\)-length permutations with a given number of left-to-right minima in in-version array representation. In this paper, we give the first Gray code for the set of \(n\)-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.
We introduce a generalization of the well-known concept of graceful labeling. Given a graph \(\Gamma\) with \(e = d.m\) edges, we define a \(d\)-graceful labeling of \(G\) as an injective function \(f: V(G) \rightarrow \{0, 1, 2, \ldots, d(m+1) – 1\}\) such that \(\{|f(x) – f(y)| : \{x, y\} \in E(\Gamma)\}\) = \(\{1, 2, 3, \ldots, d(m+1) – 1\} – \{m+1, 2(m+1), \ldots, (d-1)(m+1)\}.\) In the case of \(d = 1\) and of \(d = e\) we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively.Also, we call \(d\)-graceful \(\alpha\)-labeling of a bipartite graph \(\Gamma\) a \(d\)-graceful labeling of \(\Gamma\) with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other
one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of \(d\)-graceful \(\alpha\)-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.
Given a graph \(H\), a graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(\pi\) containing \(H\) as a subgraph.In this paper, we characterize potentially \(K_6 – E(K_3)\)-graphic sequences without zero terms, where \(K_6 – E(K_3)\) denotes the graph obtained from a complete graph on \(6\) vertices by deleting three edges forming a triangle.This characterization implies the value of \(\sigma(K_6 – E(K_3), n)\).
We propose and study game-theoretic versions of independence
in graphs. The games are played by two players – the aggressor and the
defender – taking alternate moves on a graph G with tokens located on
vertices from an independent set of \(G\). A move of the aggressor is to select
a vertex v of \(G\). A move of the defender is to move tokens located on
vertices in \(N_G(v)\) each along one incident edge. The goal of the defender is
to maintain the set of occupied vertices independent while the goal of the
aggressor is to make this impossible. We consider the maximum number of
tokens for which the aggressor can not win in a strategic and an adaptive
version of the game.
In this study, we investigate Diophantine equations using the generalized Fibonacci and Lucas sequences. We obtain all integer solutions for several Diophantine equations such as \(x^2 -kxy- y^2 = \mp 1,\) \(x^2 -kxy+ y^2 = 1,\) \(x^2 – kxy-y^2 = \mp (k^2+4),\)
\(x^2 – (k^2 + 4)xy + (k^2+4)y^2 =\mp k^2,\) \(x^2 – kxy +y^2 = -(k^2-4)\). and \(x^2-(k^2-4)xy-(k^2-4)y^2=k^2\)
Some of these results are previously known, but we provide new and distinct proofs using generalized Fibonacci and Lucas sequences.
Let \(G = \{g_1, \ldots, g_n\}\) be a finite abelian group. Consider the complete graph \(K_n\) with vertex set \(\{g_1, \ldots, g_n\}\). A \(G\)-coloring of \(K_n\) is a proper edge coloring where the color of edge \(\{g_i, g_j\}\) is \(g_i + g_j\), \(1 \leq i 2\), there exists a proper edge coloring of \(K_p\) which is decomposable into multicolored Hamilton cycles.
It is shown that \(r(K_{1,m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{m+1}\) for any two fixed integers \(k \geq m \geq 2\) and \(n \to \infty\).
This result is obtained using the analytic method and the function \(f_{m}(x) = \int_0^1 \frac{(1-t)^{\frac{1}{m}}dt}{m+(x-m)^t} , \quad x \geq 0,m \geq 1,\)
building upon the upper bounds for \(r(K_{m,k}, K_n)\) established by Y. Li and W. Zang.Furthermore, \((c – o(1)) (\frac{n}{log n})^{\frac{7}{3}}\leq r(W_{4}, K_n) \leq (1 + o(1)) (\frac{n}{log n})^{3}\) (as \(n \to \infty\)). Moreover, we derive
\(r(K_{1} + K_{m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{l+m}\) for any two fixed integers \(k \geq m \geq 2\) (as \(n \to \infty\)).
A simple graph \(G = (V, E)\) admits an \(H\)-covering if every edge in \(E\) belongs to a subgraph of \(G\) isomorphic to \(H\). We say that \(G\) is \(H\)-magic if there exists a total labeling \(f: V \cup E \rightarrow \{1, 2, \ldots, |V| + |E| + 1\}\) such that for each subgraph \(H’ = (V’, E”)\) of \(G\) isomorphic to \(H\),
\(\sum_{v \in V’} f(v) + \sum_{e \in E”} f(e)\)
is constant.
When \(f(V) = \{1, 2, \ldots, |V|\}\), then \(G\) is said to be \(H\)-supermagic.
In this paper, we show that all prism graphs \(C_n \times P_m\), except for \(n = 4\), the ladder graph \(P_3 \times P_n\), and the grid \(P_3 \times P_n\), are \(C_4\)-supermagic.
The average crosscap number of a graph \(G\) is the expected value of the crosscap number random variable, over all labeled \(2\)-cell non-orientable embeddings of \(G\). In this study, some experimental results for average crosscap number are obtained. We calculate all average crosscap numbers of graphs with Betti number less than \(5\). As a special case, the smallest ten values of average crosscap number are determined. The distribution of average crosscap numbers of all graphs in \({R}\) is sparse. Some structure theorems for average crosscap number with a given or bounded value are provided. The exact values of average crosscap numbers of cacti and necklaces are determined. The crosscap number distributions of cacti and necklaces of type \((r,0)\) are proved to be strongly unimodal, and the mode of the embedding distribution sequence is upper-rounding or lower-rounding of its average crosscap number. Some open problems are also proposed.
A Roman dominating function of a graph \(G\) is a labeling \(f: V(G) \rightarrow \{0,1,2\}\) such that every vertex with label \(0\) has a neighbor with label \(2\). The Roman domination number \(\gamma_R(G)\) of \(G\) is the minimum of \(\sum_{v \in V(G)} f(v)\) over such functions. The Roman domination subdivision number \(sd_{\gamma R}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the Roman domination number.
In this paper, we prove that if \(G\) is a graph of order \(n \geq 4\) such that \(\overline{G}\) and \(G\) have connected components of order at least \(3\), then
\(sd_{\gamma R}(G) + sd_{\gamma R}(\overline{G}) \leq \left\lfloor \frac{n}{2} \right\rfloor + 3.\)
In \textit{Ars Comb.} \({84} (2007), 85-96\), Pedersen and Vestergaard posed the problem of determining a lower bound for the number of independent sets in a tree of fixed order and diameter \(d\). Asymptotically, we give here a complete solution for trees of diameter \(d \leq 5\). The lower bound is \(5^{\frac{n}{3}}\) and we give the structure of the extremal trees. A generalization to connected graphs is stated.
Let \(\mathcal{G}\) be a family of graphs. The anti-Ramsey number \(\text{AR}(n, \mathcal{G})\) for \(\mathcal{G}\) is the maximum number
of colors in an edge coloring of \(K_n\) that has no rainbow copy of
any graph in \(\mathcal{G}\). In this paper, we determine the bipartite anti-Ramsey number for the family of trees with
\(k\) edges.
Let \(G\) be a finite group of order \(n\) and \(S\) (possibly containing the identity element) be a subset of \(G\). The Bi-Cayley graph
\(\text{BC}(G, S)\) of \(G\) is a bipartite graph with vertex set \(G \times \{0, 1\}\) and edge set \(\{(g, 0), (gs, 1) \mid g \in G, s \in S\}\). Let \(p\) (\(0 < p < 1\)) be a fixed number.We define \({B} = \{\text{BC}(G, S) \mid S \subseteq G\}\)
as a sample space and assign a probability measure by requiring \(P_r(X) = p^k q^{n-k}\) for \(X = \text{BC}(G, S)\) with \(|S| = k\),
where \(q = 1-p\). It is shown that the probability of the set of Bi-Cayley graphs of \(G\) with diameter \(3\) approaches \(1\) as the order \(n\) of \(G\) approaches infinity.
In this study, we define and investigate the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers. We derive generating functions, Binet formulas, explicit formulas, and matrix representations for these numbers. Additionally, we present explicit combinatorial and determinantal expressions, examine negatively subscripted numbers, and establish various identities. Our results parallel those for the Jacobsthal and Jacobsthal Lucas numbers, yielding interesting consequences for the Gaussian Jacobsthal and Gaussian Jacobsthal Lucas numbers.
A signed total \(k\)-dominating function of a graph \(G = (V, E)\) is a function \(f: V \rightarrow \{+1, -1\}\) such that for every vertex \(v\), the sum of the values of \(f\) over the open neighborhood of \(v\) is at least \(k\). A signed total \(k\)-dominating function \(f\) is minimal if there does not exist a signed total \(k\)-dominating function \(g\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\).The weight of a signed total \(k\)-dominating function is \(w(f) = \sum_{v \in V} f(v)\). The signed total \(k\)-domination number of \(G\), denoted by \(\gamma_{t,k}^s(G)\), is the minimum weight of a signed total \(k\)-dominating function on \(G\).The upper signed total \(k\)-domination number \(\Gamma_{t,k}^s(G)\) of \(G\) is the maximum weight of a minimal signed total \(k\)-dominating function on \(G\).
In this paper, we present sharp lower bounds on \(\gamma_{t,k}^s(G)\) for general graphs and \(K_{r+1}\)-free graphs and characterize the extremal graphs attaining some lower bounds. Also, we give a sharp upper bound on \(\Gamma_{t,k}^s(G)\) for an arbitrary graph.
We show that a \(2\)-subset-regular self-complementary \(3\)-uniform hypergraph with \(7\) vertices exists if and only if \(n \geq 6\) and \(n\) is congruent to \(2\) modulo \(4\).
Given a graph \(G\), a function \(f: V(G) \to \{1, 2, \ldots, k\}\) is a \(k\)-ranking of \(G\) if \(f(u) = f(v)\) implies every \(u-v\)
path contains a vertex \(w\) such that \(f(w) > f(u)\). A \(k\)-ranking is minimal if the reduction of any label greater
than \(1\) violates the described ranking property.The \(arank\) number of a graph, denoted \(\psi_r(G)\),
is the maximum \(k\) such that \(G\) has a minimal \(k\)-ranking.We establish new properties for minimal rankings and present
new results for the \(arank\) number of a cycle.
In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs \(G_1\) and \(G_2\) are equal to \(\kappa_1 v_2\) and \(\min\{\lambda_1 v_2^2, \delta_2 + \delta_1v_2\}\), respectively, where \(\delta_i\), \(\kappa_i\), \(\lambda_i\), and \(n_i\) denote the minimum degree, connectivity, edge-connectivity, and number of vertices of \(G_i\), respectively.
We also obtain that the edge-connectivity of the direct product of \(K_2\) and a graph \(H\) is equal to \(\min\{2\lambda, 2\beta, \min_{j =\lambda}^\delta\{j + 2\beta_j\}\}\), where \(\theta\) is the minimum size of a subset \(F \subset E(H)\) such that \(H – F\) is bipartite and \(\beta_j = \min\{\beta(C)\}\), where \(C\) takes over all components of \(H – B\) for all edge-cuts \(B\) of size \(j \geq \lambda=\lambda (H)\).
The induced path number \( \rho(G) \) of a graph \( G \) is defined as the minimum number of subsets into which the vertex set of \( G \) can be partitioned so that each subset induces a path. A Nordhaus-Gaddum type result is a (tight) lower or upper bound on the sum (or product) of a parameter of a graph and its complement. If \( G \) is a subgraph of \( H \), then the graph \( H – E(G) \) is the complement of \( G \) relative to \( H \). In this paper, we consider Nordhaus-Gaddum type results for the parameter \( \rho \) when the relative complement is taken with respect to the complete bipartite graph \( K_{m,n} \).
Rado constructed a (simple) denumerable graph \( R \) with the positive integers as vertex set with the following edges: For given \( m \) and \( n \) with \( m < n \), \( m \) is adjacent to \( n \) if \( n \) has a \( 1 \) in the \( m \)'th position of its binary expansion. It is well known that \( R \) is a universal graph in the set \( \mathcal{I} \) of all countable graphs (since every graph in \( \mathcal{I} \) is isomorphic to an induced subgraph of \( R \)) and that \( R \) can be characterized using this notion and that of being homogeneous and having the extension property. In this paper, we extend these notions to arbitrary induced-hereditary properties (of graphs), relate them to the construction of a universal graph for any such property, and obtain results which remind one of some characterizations of \( R \).
In this note, we prove that for any tree \( T \), \( \gamma_{\leq2}(T) \leq \gamma_\gamma(T) \leq ir(T) \leq \gamma(T) \), where \( \gamma_{\leq2}(G) \) is the distance-2 domination number, \( ir(T) \) is the (lower) irredundance number, \( \gamma(T) \) is the domination number, and \( \gamma_\gamma(T) \), newly defined here, equals the minimum cardinality of a set of vertices that dominates a minimum dominating set of \( T \).
A graph is \((k, l)\)-colorable if its vertex set can be partitioned into \( k \) independent sets and \( l \) cliques. A graph is chordal if it does not contain any induced cycle of length at least four. A theorem by Hell et al. states that a chordal graph is \((k, l)\)-colorable if and only if it does not contain \((l+1)K_{k+1}\) as an induced subgraph. Presented here is a short alternative proof of this result, using the characterization of chordal graphs via perfect elimination orderings.
A subset \( X \) of the vertex set of a graph \( G \) is a secure dominating set of \( G \) if \( X \) is a dominating set of \( G \) and if, for each vertex \( u \) not in \( X \), there is a neighboring vertex \( v \) of \( u \) in \( X \) such that the swap set \( (X – \{v\}) \cup \{u\} \) is again a dominating set of \( G \). The secure domination number of \( G \), denoted by \( \gamma_s(G) \), is the cardinality of a smallest secure dominating set of \( G \). In this paper, we present two algorithms (a branch-and-reduce algorithm as well as a branch-and-bound algorithm) for determining the secure domination number of a general graph \( G \) of order \( n \). The worst-case time complexities of both algorithms are \( \mathcal{O}(2^{n-s-\sum_{i=1}^{k}(|\mathcal{R}_i|-1)}) \), where \( s \) is the number of support vertices in \( G \) and \( \mathcal{R}_i, \ldots, \mathcal{R}_k \) are the redundancy classes of \( G \) (two vertices are in the same redundancy class if they are adjacent and share the same closed neighborhood which forms a clique in \( G \)).
The distinguishing chromatic number of a graph \( G \) is the least integer, \( \chi_D(G) \), for which \( G \) has a coloring of its vertices so that adjacent vertices receive different colors, and the identity is the only automorphism of \( G \) that preserves vertex colors. Our focus is on determining the distinguishing chromatic numbers of wreath products of graphs, extending the work of Tang. We prove that if \( C_n \) is a cycle with \( n \) vertices and \( P_n \) is a path with \( n \) vertices, then \( \chi_D(C_n[G]) \) and \( \chi_D(P_n[G]) \) can be found for any connected graph \( G \). We also obtain an upper bound on \( \chi_D(T[G]) \) when \( T \) is a tree and \( G \) is any connected graph. Some of our results depend on the notion of inequivalent colorings. Cheng introduces inequivalent colorings and provides a formula for computing the number of inequivalent distinguishing \( k \)-colorings of a rooted tree. We add to this work by obtaining an expression for computing the number of inequivalent distinguishing \( k \)-colorings of a cycle.
A graph \( G \) is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a quadrilateral is called a (planar) quadrangulation. In the present paper, we characterize those planar quadrangulations which are well-covered.
Suppose \( V \) is a finite set and \( \mathcal{C} \) a collection of subsets of \( V \) that contains \( \emptyset \) and \( V \) and is closed under taking intersections. Then the ordered pair \( (V, \mathcal{C}) \) is called a \({convexity}\) and the elements of \( \mathcal{C} \) are referred to as \({convex\; sets}\). For a set \( S \subseteq V \), the \({convex\; hull}\) of \( S \) relative to \( \mathcal{C} \), denoted by \( CH_{\mathcal{C}}(S) \), is the smallest convex set containing \( S \). The \({Carathéodory\; number}\), relative to a given convexity, is the smallest integer \( c \) such that for any subset \( S \) of \( V \) and any point \( v \in CH_{\mathcal{C}}(S) \), there is a subset \( F \) of \( S \) with \( |F| \leq c \) such that \( v \in CH_{\mathcal{C}}(F) \). A subset \( X \) of \( V \) is said to admit a \({Radon \;partition}\) if \( X \) can be partitioned into two sets \( X_1 \) and \( X_2 \) such that \( CH_{\mathcal{C}}(X_1) \cap CH_{\mathcal{C}}(X_2) \neq \emptyset \). The \({Radon\; number}\) of a convexity is the smallest integer \( r \) (if it exists) such that every subset \( X \) of \( V \) with at least \( r \) elements admits a Radon partition.
A set \( S \) of vertices in a graph \( G \) with vertex set \( V \) is \({digitally}\) convex if for every vertex \( v \in V \), \( N[v] \subseteq N[S] \) implies \( v \in S \). A set \( X \) is \({irredundant}\) if \( N[X] – N[X – \{x\}] \neq \emptyset \) for all \( x \in X \). The maximum cardinality of an irredundant set is the \({upper\; irredundance\; number}\) of \( G \), denoted by \( IR(G) \). A set \( X \) of vertices in a graph \( G \) is a \({local\; irredundant}\) set for a vertex \( v \) of \( G \), if for each \( x \in X \), \( x \in N[v] – N[X – \{x\}] \) or \( x \) is adjacent to a vertex of \( N[v] – N[X – \{x\}] \). The \({upper\; local \;irredundance \;number}\) of \( v \), denoted by \( l_{IR}(v) \), is the maximum cardinality of a local irredundant set for \( v \). The \({upper\; local\; irredundance\; number}\) of a graph \( G \), denoted by \( l_{IR}(G) \), is defined as \( l_{IR}(G) = \max \{ l_{IR}(v) \mid v \in V \} \).
We show that for the digital convexity of a graph \( G \):
(i) The Carathéodory number equals \( l_{IR}(G) \).
(ii) The Radon number is bounded above by \( IR(G) + 1 \) and below by \( \beta(G) + 1 \) where \( \beta(G) \) is the independence number of \( G \). For the latter result, it is shown that there are classes of graphs for which the lower (respectively, upper) bound is attained, while the difference between the upper irredundance number and the independence number can be made as large as we wish. Moreover, there are graphs for which the Radon number of the digital convexity lies strictly between the bounds given in (ii) and does not equal one more than the upper domination number.
In this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in the case of trees, and seems to represent a unifying theory within the context of the support of the null space. We extend this idea and apply it to describe the null vectors and corresponding nullities of certain symmetric matrices associated with cycles.
A set of vertices in a graph \( G \) is a global dominating set of \( G \) if it dominates both \( G \) and its complement \( \overline{G} \). The minimum cardinality of a global dominating set of \( G \) is the global domination number of \( G \). We explore the effects of graph modifications (edge removal, vertex removal, and edge addition) on the global domination number. In particular, for each graph modification, we study the global domination stable trees, that is, the trees whose global domination number remains the same upon the modification. We characterize these stable trees having small global domination numbers.
A digraph is called \({homogeneous}\) if every connected induced sub-digraph with two or more vertices is either strong or acyclic. The class of homogeneous digraphs contains acyclic digraphs, round digraphs, and symmetric digraphs. Tournaments which are homogeneous have been studied by Guido, Moon, and others, and characterized by Moon. In this paper, we give a characterization of homogeneous digraphs. Our characterization reveals a nice structural property of this class of digraphs and shows that all homogeneous digraphs can be obtained from acyclic digraphs, round digraphs, and symmetric digraphs by the operation of substitution.
We analyze TIMBER, a game played on graphs. We find the \(\mathcal{P}\) positions for both normal and misère play on paths and show how to win the game. In passing, we also show a correspondence with Dyck paths, the Catalan, and Fine numbers. We present an algorithm for winning the Normal Play game on trees.
An edge ordering of a graph \( G \) is an injection \( f : E(G) \to \mathbb{Z} \), where \( \mathbb{Z} \) denotes the set of integers. A path in \( G \) for which the edge ordering \( f \) increases along its edge sequence is called an \( f \)-\({ascent}\); an \( f \)-ascent is maximal if it is not contained in a longer \( f \)-ascent. The \({depression}\) of \( G \) is the smallest integer \( k \) such that any edge ordering \( f \) has a maximal \( f \)-ascent of length at most \( k \). We apply the concept of ascents to edge colorings using possibly less than \( |E(G)| \) colors and consider the problem of determining the minimum number of colors required such that there exists an edge coloring \( c \) for which the length of a shortest maximal \( c \)-ascent is equal to the depression of \( G \).
An independent set of a graph \( G \) is a set of vertices of \( G \) which are pairwise non-adjacent. There are many applications for which the input is a graph \( G \) with a large symmetry group and the goal is to generate up to isomorphism all of the independent sets, all of the maximal independent sets, or all of the maximum independent sets. This paper presents a very fast practical algorithm for these problems. The tactic can also be applied to many other problems: some examples are generation of all dominating sets, colorings, or matchings of a graph up to isomorphism.
A graph \( G \) is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the labels. The minimum size of a label class in such a labeling of \( G \) is called the cost of 2-distinguishing and is denoted by \( \rho(G) \). This paper shows that \( \rho(K_{2^m-1}:2^{m-1}-1) = m+1 \) — the only result so far on the cost of 2-distinguishing Kneser graphs. The result for Kneser graphs is adapted to show that \( \rho(Q_{2^m-2}) = \rho(Q_{2^m-1}) = \rho(Q_{2^m}) = m+2 \) — a significant improvement on previously known bounds for the cost of 2-distinguishing hypercubes.
We explore cops-and-robbers games in several directions, giving partial results in each and refuting two reasonable conjectures. We close with some open problems.
Given a set \( S \subseteq V \) in a graph \( G = (V, E) \), we say that a vertex \( v \in V \) is perfect if \( |N[v] \cap S| = 1 \), that is, the closed neighborhood \( N[v] = \{v\} \cup \{u \mid uv \in E\} \) of \( v \) contains exactly one vertex in \( S \). A vertex \( v \) is almost perfect if it is either perfect or is adjacent to a perfect vertex. Similarly, we can say that a set \( S \subset V \) is (almost) perfect if every vertex \( v \in S \) is (almost) perfect; \( S \) is externally (almost) perfect if every vertex \( u \in V – S \) is (almost) perfect; and \( S \) is completely (almost) perfect if every vertex \( v \in V \) is (almost) perfect. In this paper, we relate these concepts of perfection to independent sets, dominating sets, efficient and perfect dominating sets, distance-2 dominating sets, and to perfect neighborhood sets in graphs. The concept of a set being almost perfect also provides an equivalent definition of irredundance in graphs.
A graph \( G \) is \( k \)-edge-\( i \)-critical if it has independent domination number \( i(G) = k \), and \( i(G + xy) < i(G) \) whenever \( xy \notin E(G) \). The following results are obtained for \( 3 \)-edge-\( i \)-critical graphs \( G \):
The proofs of these results rely on a closure operation, a characterization of the \( 2 \)-connected, \( 3 \)-edge-\( i \)-critical graphs with \( \delta = 2 \), and a characterization of the \( 3 \)-edge-\( i \)-critical graphs with a cut vertex.
An identifying code in a graph \( G \) is a set \( D \) of vertices such that the closed neighborhood of each vertex of the graph has a nonempty, distinct intersection with \( D \). The minimum cardinality of an identifying code is denoted \( \gamma^{ID}(G) \). Building upon recent results of Gravier, Moncel, and Semri, we show for \( n \leq m \) that \( \gamma^{ID} (K_n \Box K_m) = \max\{2m – n, m + \lfloor n/2 \rfloor\} \). Furthermore, we improve upon the bounds for \( \gamma^{ID}(G \Box K_m) \) and explore the specific case when \( G \) is the Cartesian product of multiple cliques.
Mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on its vertices. The locations of the guards must induce a vertex cover at all times. We compare this new model of graph protection with other previously studied parameters, including such as the eternal domination number and the variation of the eternal vertex cover problem in which attacks occur at edges
Suppose each vertex in a graph \( G \) has a unit of information and that all the units must be collected at a vertex \( u \) in \( G \). Assuming that a vertex can receive (from its neighbors) an unlimited number of units at each discrete moment but can only send one at a time, find the shortest collection time, \( \operatorname{col}_u(G) \), needed to collect all the information at \( u \) and an optimal protocol that achieves this.
We derive lower and upper bounds for the problem, give a polynomial time algorithm in the general case, and a linear time algorithm for hypercubes.
A (di)graph \( G \) is \({homomorphically; full}\) if every homomorphic image of \( G \) is a sub(di)graph of \( G \). This class of (di)graphs arose in the study of whether a homomorphism from a given graph \( G \) to a fixed graph \( H \) can be factored through a fixed graph \( Y \). Brewster and MacGillivray proved that the homomorphically full irreflexive graphs are precisely the graphs that contain neither \( P_4 \) nor \( 2K_2 \) as an induced subgraph. In this paper, we show that the homomorphically full reflexive graphs are precisely threshold graphs, i.e., the graphs that contain none of \( P_4 \), \( 2K_2 \), and \( C_4 \) as an induced subgraph. We also characterize the reflexive semicomplete digraphs that are homomorphically full, and discuss the relationship of these digraphs and Ferrers digraphs.
The eternal domination number of graph \( G \) is the smallest set of mobile guards which can defend \( G \) against an infinite sequence of attacks on its vertices. In this paper we give results for the eternal domination numbers of \( P_4 \Box P_n \).
A red-blue coloring of a graph \( G \) is an edge coloring of \( G \) in which every edge of \( G \) is colored red or blue. Let \( F \) be a connected graph of size 2 or more with a red-blue coloring, at least one edge of each color, where some blue edge of \( F \) is designated as the root of \( F \). Such an edge-colored graph \( F \) is called a color frame. An \( F \)-coloring of a graph \( G \) is a red-blue coloring of \( G \) in which every blue edge of \( G \) is the root edge of a copy of \( F \) in \( G \). The \( F \)-chromatic index \( \chi’_F(G) \) of \( G \) is the minimum number of red edges in an \( F \)-coloring of \( G \). It has been shown that these concepts generalize both edge domination and matchings in graphs. In this paper, we consider the two color frames \( Y_1 \) and \( Y_2 \) that result from the claw \( K_{1,3} \), where \( Y_1 \) has exactly one red edge and \( Y_2 \) has exactly two red edges. An edge \( e \) in a graph \( G \) is a non-claw edge if \( e \) belongs to no claw in \( G \). It is shown that if \( G \) is a connected graph containing \( \ell \) non-claw edges, then \( \chi’_{Y_1}(G) \leq \chi’_{Y_2}(G) \leq 3\chi’_{Y_1}(G) – 2\ell \) and \( \chi’_{Y_1}(G) = \chi’_{Y_2}(G) \) if and only if \( G \) is a path or cycle. Furthermore, a pair \( a, b \) of positive integers can be realized as the \( Y_1 \)-chromatic index and \( Y_2 \)-chromatic index for some connected graph of order at least 4 if and only if \( a \leq b \leq 3a \) and \( b \geq 2 \).
Consider an n-set, say \(X_n = {1,2,…,n}\). An exponential generating function and recurrence relation for the number of subpermutations of \(X_n\), whose orbits are of size at most \(k \geq 0\) are obtained. Similar results for
the number of nilpotent subpermutations of nilpotency index at most \(k\), and exactly \k\) are also given, along with arithmetic and asypmtotic formulas for these numbers. \(1\) \(2\)
In this paper, we show that the crossing number of the complete tripartite graph \(K_{2,4,n}\) is \(6\left\lfloor\frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor+2n\).
An \((n \times n)\) matrix \(A = (a_{ij})\) is called a Toeplitz matrix
if it has constant values along all diagonals parallel to the main diagonal.
A directed Toeplitz graph is a digraph with Toeplitz adjacency matrix.
In this paper, we discuss conditions for the existence of Hamiltonian cycles
in directed Toeplitz graphs.
For \(n \geq 2\) and a local field \(K\), let \(\Delta_n\) denote the affine building naturally associated to the symplectic group \(\mathrm{Sp}_{n}(K)\). We compute the spectral radius of the subgraph \(Y_n\) of \(\Delta_n\) induced by the special vertices in \(\Delta_n\), from which it follows that \(Y_n\) is an analogue of a family of expanders and is non-amenable.
The concept of \(t\)-(v, \(\lambda\)) trades of block designs has been studied in detail. See, for example, A.~S. Hedayat (1990) and Billington (2003). Latin trades have also been extensively studied under various names; see A.~D. Keedwell (2004) for a survey. Recently, Khanban, Mahdian, and Mahmoodian have extended the concept of Latin trades and introduced \(t\)-(\(v, k\)) Latin trades.In this paper, we study the spectrum of possible volumes of these trades, \(S(t, k)\). Firstly, similarly to trades of block designs, we consider \((t+2)\) numbers \(s_i = 2^{i+1}-2^{(t+1)-i} \), \(0 \leq i \leq t+1\), as critical points. Then, we show that \(s_i \in S(t,k)\) for any \(0 \leq i \leq t+1\), and if \(s \in (s_i, s_{i+1}, )\), \(0 \leq i \leq t\), then \(s \notin S(t, t+1)\). As an example, we precisely determine \(S(3, 4)\).
This paper investigates the relationship between the degree-sum of adjacent vertices, girth, and upper embeddability of graphs, combining it with edge-connectivity. The main result is:
Let \(G\) be a \(k\)-edge-connected simple graph with girth \(g\). If there exists an integer \(m\) (\(1 \leq m \leq g\)) such that for any \(m\) consecutively adjacent vertices \(x_i\) (\(i = 1, 2, \ldots, m\)) in any non-chord cycle \(C\) of \(G\), it holds that
\[\sum\limits_{i=1}^m d_G(x_i) > \frac{mn}{(k-1)^2+2} + \frac{km}{g}+(2-g)m,\]
where \(k = 1, 2, 3, n = |V(G)|\), then \(G\) is upper embeddable and the upper bound is best possible.
In this study, we define and investigate the Bivariate Gaussian Fibonacci and Bivariate Gaussian Lucas Polynomials. We derive generating functions, Binet formulas, explicit formulas, and partial derivatives of these polynomials. By defining these bivariate polynomials for special cases, we obtain:\(F_n(x, 1)\) as the Gaussian Fibonacci polynomials,\(L_n(x, 1)\) is the Gaussian Lucas polynomials,\( {F}_{n}(1, 1)\) as the Gaussian Fibonacci numbers, and \( {L}_{n}(1, 1)\) as the Gaussian Lucas numbers, as defined in \([19]\).
In this paper, we show that the set \(\{E_0(x), E_1(x), \ldots, E_n(x)\}\) of Euler polynomials is a basis for the space of polynomials of degree less than or equal to \(n\). From the properties of Euler basis polynomials, we derive some interesting identities on the product of two Bernoulli and Euler polynomials.
An \(n\)-colour even composition is defined as an \(n\)-colour composition with even parts. In this paper, we obtain generating functions, explicit formulas, and a recurrence formula for \(n\)-colour even compositions.
In this paper, we characterize boundedness and compactness of products of composition operators induced by the lens and the lunar maps and iterated differentiation acting between Hardy and weighted Bergman spaces of the unit disk in terms of the angle of contact of these maps with the unit circle.
Let \(G = (V(G), E(G))\) be a graph and \(\alpha(G)\) be the independence number of \(G\). For a vertex \(v \in V(G)\), \(d(v)\) and \(N(v)\) represent the degree and the neighborhood of \(v\) in \(G\), respectively.In this paper, we prove that if \(G\) is a \(k\)-connected graph of order \(n\), where (\(k \geq 2\)) graph of order \(n\) and \(\max\{d(v) : v \in S\} \geq \frac{n}{2}\) for every independent set \(S\) of \(G\) with \(|S| = k\) which has two distinct vertices \(x, y \in S\) satisfying \(1\leq |N(x) \cap N(y)| \leq \alpha(G) – 2,\)
then either \(G\) is hamiltonian or else \(G\) belongs to one of a family of exceptional graphs.We also establish a similar sufficient condition for Hamiltonian-connected graphs.
In this paper, we generalize the companion Pell sequence. We provide combinatorial, graph, and matrix representations of this sequence.Using these representations, we describe some properties of the generalized Pell numbers and the generalized companion Pell numbers. We define the golden Pell matrix for determining the generalized Pell sequences and, among other results, prove the “generalized Cassini formula” for them.Moreover, we establish some relations between generalized Pell numbers and the classical Fibonacci numbers.
In this paper, we determine the third largest and the fourth largest numbers of independent sets among all trees of order \(n\). Moreover, we determine the \(k\)-th largest numbers of independent sets among all forests of order \(n\), where \(k \geq 2\). Besides, we characterize those extremal graphs achieving these values.
For a set \(\mathcal{P}\) of permutations, the sign-imbalance of \(\mathcal{P}\) is the difference between the numbers of even and odd permutations in \(\mathcal{P}\).In this paper, we determine the sign-imbalances of two classes of alternating permutations ,one is the Alternating permutations avoiding a pattern of length three and the other is the Alternating permutations of genus \(0\)
The sign-imbalance of the former involves Catalan and Fine numbers, and that of the latter is always \(\pm 1\).Meanwhile, we give a simpler proof of Dulucq and Simion’s result on the number of alternating permutations of genus \(0\).
A survivable path \((W, P)\) between a pair of vertices \(x_i, x_j\) in an undirected simple graph \(G\) is an ordered pair of edge-disjoint simple paths consisting of a working path \(W = x_i, \ldots, x_j\) a protection path \(P = x_i, \ldots, x_j\).An optimal set of survivable paths in graph \(G\) corresponds to a set of mesh-restored lightpaths defined on an optical network that minimizes the number of used optical channels.In this paper, we present new properties of the working paths, which are contained in an optimal set of survivable paths in \(G\).
We describe the global behavior of the nonnegative equilibrium points of the difference equation
\[x_{n+1} = \frac{ax_{n -p}}{b+c \prod\limits_{i=0}^{k} x_{n-(2i+1)}},n=0,1,\ldots,\]
where \(k,p \in \mathbb{N}\), parameters \(a,b,c\) and initial conditions are nonnegative real numbers.
Let \(\mathcal{T}_{n,n-4}\) be the set of trees on \(n\) vertices with diameter \(n-4\). In this paper, we determine the unique tree which has the minimal Laplacian spectral radius among all trees in \(\mathcal{T}_{n,n-4}\).
This work is related to that of Yuan [The minimal spectral radius of graphs of order n with diameter \(n – 4\), Linear Algebra Appl. \(428(2008)2840-2851]\), which determined the graph with minimal spectral radius among all the graphs of order \(n\) with diameter \(n-4\). We can observe that the extremal tree on the Laplacian spectral radius is different from that on the spectral radius.
We introduce the notion of vague Lie sub-superalgebras (resp. vague ideals) and present some of their properties. We investigate the properties of vague Lie sub-superalgebras and vague ideals under homomorphisms of Lie superalgebras.We introduce the concept of vague bracket product and establish its characterizations. We also introduce the notions of solvable vague ideals and nilpotent vague ideals of Lie superalgebras and present the corresponding theorems parallel to Lie superalgebras.
The atom-bond connectivity (ABC) index of a graph \(G\) is defined in mathematical chemistry as\(\mathrm{ABC}(G) = \sum_{uv \in E(G)} \sqrt{\frac{d_u +d_v-2}{ d_u d_v}},\) where \(E(G)\) is the edge set of \(G\) and \(d_u\) is the degree of vertex \(u\) in \(G\).In this paper, we determine the unique graphs with the largest and the second largest ABC indices, respectively, in the class of unicyclic graphs on \(2m\) vertices with perfect matchings.
Let \(\Delta\) be one of the dual polar spaces \(\mathrm{DQ}(8, q)\), \(\mathrm{DQ}^-(7,q)\), and let \(e: \Delta \to \Sigma\) denote the spin-embedding of \(\Delta\). We show that \(e(\Delta)\) is a two-intersection set of the projective space \(\Sigma\). Moreover, if \(\Delta \cong \mathrm{DQ}^-(7,q)\), then \(e(\Delta)\) is a \((q^3 + 1)\)-tight set of a nonsingular hyperbolic quadric \(\mathrm{Q}^+(7,q^2)\) of \(\Sigma \cong PG(7,q^2)\). This \((q^2 + 1)\)-tight set gives rise to more examples of \((q^3 + 1)\)-tight sets of hyperbolic quadrics by a procedure called field-reduction.All the above examples of two-intersection sets and \((q^3 + 1)\)-tight sets give rise to two-weight codes and strongly regular graphs.
Let \(G = (V, E)\) be a simple undirected graph. An independent set is a subset \(S \subseteq V\) such that no two vertices in \(S\) are adjacent. A maximal independent set is an independent set that is not a proper subset of any other independent set.
In this paper, we study the problem of determining the fourth largest number of maximal independent sets among all trees and forests. Extremal graphs achieving these values are also given.
From differential operators and the generating functions of Bernoulli and Euler polynomials, we derive some new theorems on Bernoulli and Euler numbers. By using integral formulae and arithmetical properties relating to the Bernoulli and Euler polynomials, we obtain new identities on Bernoulli and Euler numbers. Finally, we give some new properties on Bernoulli and Euler numbers arising from the \(p\)-adic integrals on \(\mathbb{Z}_p\).
Let \(u,v\) be two vertices of a connected graph \(G\). The vertex \(v\) is said to be a boundary vertex of \(u\) if no neighbor of \(v\) is further away from \(u\) than \(v\). The boundary of a graph is the set of all its boundary vertices.In this work, we present a number of properties of the boundary of a graph under different points of view:(1) A realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary.(2) The contour is a monophonic set.(3) The cardinality of the boundary is an upper bound for both the metric dimension and the determining number of a graph.
Computing the crossing number of a given graph is, in general, an elusive problem, and only the crossing numbers of a few families of graphs are known. Most of them are the Cartesian products of special graphs. This paper determines the crossing number of the Cartesian product of a 6-vertex graph with the star \(S_n\).
Let \(M = (E, \mathcal{F})\) be a matroid on a set \(E\), \(B\) one of its bases, and \(M_B\) the base matroid associated to \(B\). In this paper, we determine a characterization of simple binary matroids \(M\) which are not isomorphic to \(M_B\), for every base \(B\) of \(M\). We also extend to matroids some graph notions.
Let \(H\) and \(G\) be two graphs (or digraphs), where \(G\) is a subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges (or arcs) of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H, G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we obtain the existence spectra of \((ADK_{m,n}, P_3^i)\)-LGD, where \(P_3^i\) (\(i = 1,2,3\)) are the three types of oriented \(P_3\).
Let \(G\) be a graph. The zeroth-order general Randić index of a graph is defined as \(R_\alpha^0(G) = \sum_{v \in V(G)} d(v)^\alpha(v)\), where \(\alpha\) is an arbitrary real number and \(d(v)\) is the degree of the vertex \(v\) in \(G\). In this paper, we give sharp lower and upper bounds for the zeroth-order general Randić index \(R_\alpha^0(G)\) among all unicycle graphs \(G\) with \(n\) vertices and \(k\) pendant vertices.
\(n\)-ary hypergroups are a generalization of Dörnte \(n\)-ary groups and a generalization of hypergroups in the sense of Marty. In this paper, we investigate some properties of \(n\)-ary hypergroups and (commutative) fundamental relations. We determine two families \( {P}(H)\) and \( {P}_\sigma(H)\) of subsets of an \(n\)-ary hypergroup \(H\) such that two geometric spaces \((H, {P}(H))\) and \((H, {P}_\sigma(H))\) are strongly transitive. We prove that in every \(n\)-ary hypergroup, the fundamental relation \(\beta\) and the commutative fundamental relation \(\gamma\) are strongly compatible equivalence relations.
In this paper, we develop a technique that allows us to obtain new effective constructions of \(1\)-resilient Boolean functions with very good nonlinearity and autocorrelation. Our strategy to construct a \(1\)-resilient function is based on modifying a bent function by toggling some of its output bits. Two natural questions that arise in this context are: “At least how many bits and which bits in the output of a bent function need to be changed to construct a \(1\)-resilient Boolean function?” We present an algorithm that determines a minimum number of bits of a bent function that need to be changed to construct a \(1\)-resilient Boolean function. We also present a technique to compute points whose output in the bent function need to be modified to get a \(1\)-resilient function. In particular, the technique is applied up to \(14\)-variable functions, and we show that the construction provides \(1\)-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation absolute indicator values, which were not known earlier.
The noncrossing matchings with each of their blocks containing a given element are introduced and studied. The enumeration of these matchings is described through a polynomial of several variables, which is proved to satisfy a recursive formula. Results of the enumeration of noncrossing matchings with fixed points are connected with Catalan numbers.
For \(1 \leq s \leq n-3\), let \(C_n(i;i_1, v_2, \ldots, i_s)\) denote an \(n\)-cycle with consecutive vertices \(x_1, x_2, \ldots, x_n\) to which the \(s\) chords \(x_{ i}x_{i_1}, x_{i}x_{i_2}, \ldots, x_{i}x_{i_s}\) have been added. In this paper, we discuss the strongly \(c\)-harmonious problem of the graph \(C_n(i;i_1, i_2, \ldots, i_s)\).
A shell of width \(n\) is a fan \(C_n(1;3,4, \ldots, n-1)\) and a vertex with degree \(n-1\) is called apex. \(MS(n^m)\) is a graph consisting of \(m\) copies of shell of width \(n\) having a common apex. If \(m \geq 1\) is odd, then the multiple shell \(MS(n^ m)\) is harmonious.
The closed neighborhood \(N_G[e]\) of an edge \(e\) in a graph \(G\) is the set consisting of \(ev\) and of all edges having a common end-vertex with \(e\). Let \(f\) be a function on \(E(G)\), the edge set of \(G\), into the set \(\{-1, 1\}\). If \(\sum_{x\in E(G)}f(x \geq 1\) for at least \(k\) edges \(e\) of \(G\), then \(f\) is called a signed edge \(k\)-subdominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge \(k\)-subdominating functions \(f\) of \(G\), is called the signed edge \(k\)-subdomination number of \(G\) and is denoted by \(\gamma_{s,k}(G)\). In this note, we initiate the study of the signed edge \(k\)-subdomination in graphs and present some (sharp) bounds for this parameter.
A graph \(G\) with vertex set \(V\) is said to have a prime labeling if its vertices can be labeled with distinct integers \(1, 2, \ldots, |V|\) such that for every edge \(xy\) in \(E(G)\), the labels assigned to \(x\) and \(y\) are relatively prime or coprime. In this paper, we show that the Knödel graph \(W_{3,n}\) is prime for \(n \leq 130\).
Let \(G = (V, E)\) be a graph. A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V – D\) has a neighbor in \(V – D\). The cardinality of a minimum total restrained dominating set in \(G\) is the total restrained domination number of \(G\). In this paper, we define the concept of total restrained domination edge critical graphs, find a lower bound for the total restrained domination number of graphs, and constructively characterize trees having their total restrained domination numbers achieving the lower bound.
Let \(\Gamma = (X, R)\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 3\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). For \(0 \leq i \leq i+s \leq d-1\), suppose \(\Delta_i\) and \(\Delta_0\) are subspaces with diameter \(i\) and \(i+s\), respectively, and with \(\Delta_i \subseteq \Delta_0\). Let \(\mathcal{L}(i, i+s; d)\) denote the set of all subspaces \(\Delta’\) with diameters \(\geq i\) such that \(d(\Delta_0 \cap \Delta’) = \Delta_1\) and \(d(\Delta_0 + \Delta’) = d(\Delta’) + s\) in \(\Gamma$ including \(\Delta_0\). If we partial order \(\mathcal{L}(i, i+s; d)\) by ordinary inclusion (resp. reverse inclusion), then \(\mathcal{L}(i, i+s; d)\) is a poset, denoted by \(\mathcal{L}_0(i, i+s; d)\) (resp. \(\mathcal{L}_R(i, i+s; d)\)). In the present paper, we show that both \(\mathcal{L}_0(i, i+s; d)\) and \(\mathcal{L}_R(i, i+s; d)\) are atomic lattices, and classify their geometricity.
By means of the partial fraction decomposition method, we evaluate a very general determinant of formal shifted factorial fractions, which covers numerous binomial determinantal identities.
Let \(H\) be a subgroup of a finite group \(G\). The relative \(n\)-th commutativity degree, denoted as \(P_n(H,G)\), is the probability of commuting the \(n\)-th power of a random element of \(H\) with an element of \(G\). Obviously, if \(H = G\) then the relative \(n\)-th commutativity degree coincides with the \(n\)-th commutativity degree, \(P_n(G)\). The purpose of this article is to compute the explicit formula for \(P_n(G)\), where \(G\) is a 2-generator \(p\)-group of nilpotency class two. Furthermore, we observe that if we have two pairs of relative isoclinic groups, then they have equal relative \(n\)-th commutativity degree.
Let \(\varphi: M \to {C}^n\) be an \(n\)-dimensional compact Willmore Lagrangian submanifold in the Complex Euclidean Space \({C}^n\). Denote by \(S\) and \(H\) the square of the length of the second fundamental form and the mean curvature of \(M\), respectively. Let \(p\) be the non-negative function on \(M\) defined by \(p^2 = S – nH^2\). Let \(K\) and \(Q\) be the functions which assign to each point of \(M\) the infimum of the sectional curvature and Ricci curvature at the point, respectively. In this paper, we prove some integral inequalities of Simons’ type for \(n\)-dimensional compact Willmore Lagrangian submanifolds \(\varphi: M \to {C}^n\) in the Complex Euclidean Space \({C}^n\) in terms of \(p^2\), \(K\), \(Q\), and \(H\), and give some rigidity and characterization theorems.
Let \(G\) be a subgraph of \(K_n\). The graph obtained from \(G\) by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a \(T(G)\)-triple. An edge-disjoint decomposition of \(3K_n\) into copies of \(T(G)\) is called a \(T(G)\)-triple system of order \(n\). If, in each copy of \(T(G)\) in a \(T(G)\)-triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of \(G\)) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(K_n\), then the \(T(G)\)-triple system is said to be perfect. The set of positive integers \(n\) for which a perfect \(T(G)\)-triple system exists is called its spectrum. Earlier papers by authors including Billington, Lindner, Kıygıkçifi, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). Then, in our previous paper, the spectrum of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and \(i (\leq 6)\) edges was determined. In this paper, we will completely solve the spectrum problem of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and seven edges.
This paper investigates tilings of a \(2 \times n\) rectangle using vertical and horizontal dominos. It is well-known that these tilings are counted by the Fibonacci numbers. We associate a graph to each tiling by converting the corners and borders of the dominos to vertices and edges. We study the combinatorial, probabilistic, and graph-theoretic properties of the resulting “domino tiling graphs.” In particular, we prove central limit theorems for naturally occurring statistics on these graphs. Some of these results are then extended to more general tiling graphs.
We prove nonexistence of circulant weighing matrices with parameters from seven previously open entries of the updated Strassler’s table. The method of proof utilizes some modular constraints on circulant weighing matrices with multipliers.
For a connected graph \( G = (V, E) \) of order at least two, a chord of a path \( P \) is an edge joining two non-adjacent vertices of \( P \). A path \( P \) is called a monophonic path if it is a chordless path. A longest \( x-y \) monophonic path is called an \( x-y \) detour monophonic path. A set \( S \) of vertices of \( G \) is a monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) monophonic path for some elements \( x \) and \( y \) in \( S \). The minimum cardinality of a monophonic set of \( G \) is the monophonic number of \( G \), denoted by \( m(G) \). A set \( S \) of vertices of \( G \) is a detour monophonic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x-y \) detour monophonic path for some \( x \) and \( y \) in \( S \). The minimum cardinality of a detour monophonic set of \( G \) is the detour monophonic number of \( G \) and is denoted by \( dm(G) \). We determine bounds for it and characterize graphs which realize these bounds. Also, for each pair \( a, b \) of integers with \( 2 \leq a \leq b \), we prove that there is a connected graph \( G \) with \( m(G) = a \) and \( dm(G) = b \).
Fault diagnosis, testing and tolerance in large scale computer and communication systems is a topic of great interest to the computer and communications research communities. In this paper, we give a broad survey of an area called system level diagnosis initiated by Preparata, Metze and Chien. Our survey includes different models of diagnosis and related diagnosis and diagnosability algorithms. In particular, we have given a detailed view of distributed diagnosis. We believe most of these works form the foundation of the research in the emerging area of fault tolerance in a mobile environment.
For vertices \( u \) and \( v \) in a connected graph \( G = (V, E) \), the monophonic detour distance \( d_m(u, v) \) is the length of a longest \( u-v \) monophonic path in \( G \). An \( u-v \) monophonic path of length \( d(u, v) \) is an \( u-v \) monophonic detour or an \( u-v \) \( m \)-detour. The set \( I_{d_m}[u, v] \) consists of all those vertices lying on an \( u-v \) \( m \)-detour in \( G \). Given a set \( S \) of vertices of \( G \), the union of all sets \( I_{d_m}[u, v] \) for \( u, v \in S \), is denoted by \( I_{d_m}[S] \). A set \( S \) is an \( m \)-detour convex set if \( I_{d_m}[S] = S \). The \( m \)-detour convex hull \( [S]_{d_m} \) of \( S \) in \( G \) is the smallest \( m \)-detour convex set containing \( S \).
A set \( S \) of vertices of \( G \) is an \( m \)-detour set if \( I_{d_m}[S] = V \) and the minimum cardinality of an \( m \)-detour set is the \( m \)-detour number \( md(G) \) of \( G \). A set \( S \) of vertices of \( G \) is an \( m \)-detour hull set if \( [S]_{d_m} = V \) and the minimum cardinality of an \( m \)-detour hull set is the \( m \)-detour hull number \( md_h(G) \) of \( G \).
Certain general properties of these concepts are studied. Bounds for the \( m \)-detour hull number of a graph are obtained. It is proved that every two integers \( a \) and \( b \) with \( 2 \leq a \leq b \) are realizable as the \( m \)-detour hull number and the \( m \)-detour number respectively, of some graph. Graphs \( G \) of order \( n \) for which \( md_h(G) = n \) or \( md_h(G) = n-1 \) are characterized. It is proved that for each triple \( a \), \( b \), and \( k \) of positive integers with \( a < b \) and \( k \geq 3 \), there exists a connected graph \( G \) with \( rad_m(G) = a \), \( diam_m(G) = b \), and \( md_h(G) = k \).
For a connected graph \( G \) of order \( n \geq 2 \), a set \( S \) of vertices of \( G \) is a geodetic set of \( G \) if each vertex \( v \) of \( G \) lies on an \( x \)-\( y \) geodesic for some elements \( x \) and \( y \) in \( S \). The geodetic number \( g(G) \) of \( G \) is the minimum cardinality of a geodetic set of \( G \). A geodetic set of cardinality \( g(G) \) is called a \( g \)-set of \( G \).
A set \( S \) of vertices of a connected graph \( G \) is an open geodetic set of \( G \) if for each vertex \( v \) in \( G \), either \( v \) is an extreme vertex of \( G \) and \( v \in S \); or \( v \) is an internal vertex of an \( x \)-\( y \) geodesic for some \( x, y \in S \). An open geodetic set of minimum cardinality is a minimum open geodetic set, and this cardinality is the open geodetic number, \( og(G) \).
A connected open geodetic set of \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) is connected. The minimum cardinality of a connected open geodetic set of \( G \) is the connected open geodetic number of \( G \) and is denoted by \( og_c(G) \).
A total open geodetic set of a graph \( G \) is an open geodetic set \( S \) such that the subgraph \( \langle S \rangle \) induced by \( S \) contains no isolated vertices. The minimum cardinality of a total open geodetic set of \( G \) is the total open geodetic number of \( G \) and is denoted by \( og_t(G) \). A total open geodetic set of cardinality \( og_t(G) \) is called an \( og_t \)-set of \( G \).
Certain general properties satisfied by total open geodetic sets are discussed. Graphs with total open geodetic number \( 2 \) are characterized. The total open geodetic numbers of certain standard graphs are determined. It is proved that for positive integers \( r \), \( d \), and \( k \geq 4 \) with \( r \leq d \leq 2r \), there exists a connected graph of radius \( r \), diameter \( d \), and total open geodetic number \( k \). It is also proved that for the positive integers \( a \), \( b \), and \( n \) with \( 4 \leq a \leq b \leq n \), there exists a connected graph \( G \) of order \( n \) such that \( og_t(G) = a \) and \( og_c(G) = b \).
In this paper, we provide a powerful technique for the existence of Hamilton-Waterloo Problem from lower order to higher order.
In 1996, Muthusamy and Paulraja conjectured that for k ≥ 3, the Cartesian product Km□Kn has a Pk-factorization if and only if mn ≡ 0 mod k and 2(k − 1)|k(m + n − 2). Recently, Chitra and Muthusamy have partially settled this conjecture for k = 3. In this paper, it is shown that for k = 4 the above conjecture is true if (m mod 12, n mod 12) ∈ {(0, 2), (2, 0), (0, 8), (8, 0), (2, 6), (6, 2), (6, 8), (8, 6), (4, 4)}. The left over cases for k = 4 are (m mod 12, n mod 12) ∈ {(0, 5), (5, 0), (0, 11), (11, 0), (1, 4), (4, 1), (3, 8), (8, 3), (4, 7), (7, 4), (4, 10), (10, 4), (8, 9), (9, 8), (10, 10)}.
In the framework of P systems introduced by Paun (1998), the generation of rectangular arrays and hexagonal arrays has been studied in the literature. In this paper, we introduce a new P system generating a family of hexagonal array languages. We compare this new family with the existing families of hexagonal array languages.
Hypertournaments are generalizations of tournaments. We discuss the concept of scores, losing scores, total scores, and degrees in \(k\)-hypertournaments and present characterizations of sequences to be score, losing score, total score, and degree sequences of some \(k\)-hypertournaments. We further discuss stronger upper and lower bounds for scores and losing scores. We extend the concept of scores, losing scores, and degrees to bipartite hypertournaments. In the end, we list some open problems in hypertournaments.
We introduce \( k \)-ctrees, which are a natural generalization of trees. A \( k \)-ctree can be constructed by recursion as follows: Any set of \( k \) independent vertices is a \( k \)-ctree, and a \( k \)-ctree of order \( n + 1 \) is obtained by inserting an \( (n + 1) \)-th vertex, and joining it to each of any \( k \) independent vertices in a \( k \)-ctree of order \( n \). We obtain basic properties and characterizations of \( k \)-ctrees involving \( k \)-degeneracy, triangle-free properties, and number of edges. Further, we determine the conditions under which \( k \)-ctrees are line, middle, or total graphs. Finally, we pose some open problems, all of them related to the characterization of \( k \)-ctrees.
An \( (a, d) \)-edge-antimagic total labeling of a graph \( G \) with \( p \) vertices and \( q \) edges is a bijection \( f \) from the set of all vertices and edges to the set of positive integers \( \{1, 2, 3, \dots, p+q\} \) such that all the edge-weights \( w(uv) = f(u) + f(v) + f(uv) \) for \( uv \in E(G) \), form an arithmetic progression starting from \( a \) and having common difference \( d \). An \( (a, d) \)-edge-antimagic total labeling is called a super \( (a, d) \)-edge-antimagic total labeling (\((a,d)\)-SEAMT labeling) if \( f(V(G)) = \{1, 2, 3, \dots, p\} \). The graph \( F_n \), consisting of \( n \) triangles with a common vertex, is called the friendship graph. The generalized friendship graph \( F_{m_1, m_2, \dots, m_n} \) consists of \( n \) cycles of orders \( m_1 \leq m_2 \leq \dots \leq m_n \) having a common vertex. In this paper, we prove that the friendship graph \( F_{16} \) does not admit a \( (a, 2) \)-SEAMT labeling. We also investigate the existence of \( (a, d) \)-SEAMT labeling for several classes of generalized friendship graphs.
Let \( G = (V, E) \) be a connected graph with domination number \( \gamma \geq 2 \). In this paper, we discuss the construction of a visual cryptography scheme for the mindom access structure \( \Gamma_D(G) \) with a basis consisting of all \( \gamma \)-sets of \( G \). We prove that the access structure \( \Gamma_D(G) \) is a \( (2, n) \)-threshold access structure if and only if \( n \) is even and \( G = K_n – M \), where \( M \) is a perfect matching in \( K_n \). Further, the \( (k, n) \)-VCS with \( k < n \) can be realized as a \( \Gamma_D(G) \)-VCS if and only if \( k = 2 \) and \( n \) is even. We also construct \( \Gamma_D(G) \)-VCS for several classes of graphs such as complete bipartite graphs, cycles \( C_n \), and \( K_n – C_n \), and we have achieved substantial reduction in the pixel expansion when compared to the VCS constructed by using other known methods.
Let \( G = (V, E) \) be a graph of order \( n \). Let \( f: V \to \{1, 2, \dots, n\} \) be a bijection. For any vertex \( v \in V \), the neighbor sum \( \sum_{u \in N(v)} f(u) \) is called the weight of the vertex \( v \) and is denoted by \( w(v) \). If \( w(x) \neq w(y) \) for any two distinct vertices \( x \) and \( y \), then \( f \) is called a distance antimagic labeling. In this paper, we present several results on distance antimagic graphs along with open problems and conjectures.
In this paper, we focus our study on finding necessary and sufficient conditions required for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) and \( (C_m \circ \overline{K}_n)^* \). In particular, we show that the necessary conditions for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) are sufficient except when none of \( m \) or \( n \) is a multiple of \( k \). In fact, our results deduce some of the results of Ushio on \( \hat{S}_k \)-factorizations of complete bipartite and tripartite symmetric digraphs.
A bipartite \( r \)-digraph is an orientation of a bipartite multigraph that is without loops and contains at most \( r \) edges between any pair of vertices from distinct parts. In this paper, we obtain necessary and sufficient conditions for a pair of sequences of non-negative integers in non-decreasing order to be a pair of sequences of numbers, called marks (or \( r \)-scores), attached to the vertices of a bipartite \( r \)-digraph. These characterizations provide algorithms for constructing the corresponding bipartite multi-digraph.
Let \( \Gamma \) be a Cayley graph generated by a transposition tree \( T \) on \( n \) vertices. In an oft-cited paper [1] (see also (9)), it was shown that the diameter of the Cayley graph \( T \) on \( n \) vertices is bounded as
\[
\text{diam}(\Gamma) \leq \max_{\pi \in S_n} \left\{ c(\pi) -n+\sum_{i=1}^{n} dist_T(i,\pi(i)) \right\},
\]
where the maximization is over all permutations \( \pi \) in \( S_n \), \( e(\pi) \) denotes the number of cycles in \( \pi \), and \( \text{distr} \) is the distance function in \( T \). It is of interest to determine for which families of trees this inequality holds with equality. In this work, we first investigate the sharpness of this upper bound. We prove that the above inequality is sharp for all trees of maximum diameter (i.e., all paths) and for all trees of minimum diameter (i.e., all stars), but the bound can still be strict for trees that are non-extremal. We also show that a previously known inequality on the distance between vertices in some families of Cayley graphs holds with equality and we prove that for some families of graphs an algorithm related to these bounds is optimal.
Let \( G = (V, E) \) be a connected graph. Two vertices \( u \) and \( v \) are said to be distance similar if \( d(u, x) = d(v, x) \) for all \( x \in V – \{u, v\} \). A nonempty subset \( S \) of \( V \) is called a pairwise distance similar set (in short `pds-set’) if either \( |S| = 1 \) or any two vertices in \( S \) are distance similar. The maximum (minimum) cardinality of a maximal pairwise distance similar set in \( G \) is called the pairwise distance similar number (lower pairwise distance similar number) of \( G \) and is denoted by \( \Phi(G) \) (\( \Phi^-(G) \)). The maximal pds-set with maximum cardinality is called a \( \Phi \)-set of \( G \). In this paper, we initiate a study of these parameters.
A signed graph (digraph) \( \Sigma \) is an ordered triple \( (V, E, \sigma) \) (respectively, \( (V, \mathcal{A}, \sigma) \)), where \( |\Sigma| := (V, E) \) (respectively, \( (V, \mathcal{A}) \)) is a graph (digraph), called the underlying graph (underlying digraph) of \( \Sigma \), and \( \sigma \) is a function that assigns to each edge (arc) of \( |\Sigma| \) a weight \( +1 \) or \( -1 \). Any edge (arc) \( e \) of \( \Sigma \) is said to be positive or negative according to whether \( \sigma(e) = +1 \) or \( \sigma(e) = -1 \). A subset \( D \subseteq V \) of vertices of \( \Sigma \) is an absorbent (respectively, a dominating set) of \( \Sigma \) if there exists a marking \( \mu: V \to \{+1, -1\} \) of \( \Sigma \) such that every vertex \( u \) of \( \Sigma \) is either in \( D \) or
\[
O(u) \cap D \neq \emptyset \quad \text{and} \quad \sigma(u, v) = \mu(u) \mu(v) \quad \forall \quad v \in O(u) \cap D,
\]
(respectively,
\[
I(u) \cap D \neq \emptyset \quad \text{and} \quad \sigma(u, v) = \mu(u) \mu(v) \quad \forall \quad v \in I(u) \cap D),
\]
where \( O(u) \) (\( I(u) \)) denotes the set of vertices \( v \) of \( \Sigma \) that are joined by the outgoing arcs \( (u, v) \) from \( u \) (incoming arcs \( (v, u) \) at \( u \)). Further, an absorbent (dominating set) of \( \Sigma \) that is independent is called a kernel (solution) of \( \Gamma \). The main aim of this paper is to initiate a study of absorbents and dominating sets in a signed graph (signed digraph), extending the existing studies on these special sets of vertices in a graph (digraph).
We recall from [13] a shell graph of size \(n\), denoted \(C(n, n-3)\), is the graph obtained from the cycle \(C_n(v_1, v_2, \ldots, v_{n-1})\) by adding \(n-3\) consecutive chords incident at a common vertex, say \(v_0\). The vertex \(v_0\) of \(C(n, n-3)\) is called the apex of the shell \(C(n, n-3)\). The vertex \(v_1\) of \(C(n, n-3)\) is said to be at level 1.
A graph \(C(2n,n-2)\) is called an alternate shell, if \(C(2n,n-2)\) is obtained from the cycle \(C_{2n}(v_0,v_1, v_2, \ldots, v_{2n-1})\) by adding \(n-2\) chords between the vertex \(v_0\) and the vertices \(v_{2i+1}\), for \(1\leq i \leq n-2\). If the vertex \(v_i\) of \(C(2n,n-2)\) at level 1 is adjacent with \(v_0\), then \(v_1\) is said to be at level 1 with a chord, otherwise the vertex \(v_1\) is said to be at level 1 without a chord.
In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph \(D\), which is the smallest positive integer \(k\) such that for every pair of vertices \(u\) and \(v\), there is a vertex \(w\) such that we can get to \(w\) from \(u\) and \(v\) in \(D\) by walks of length \(k\). In this paper, we study and obtain the scrambling indices of all primitive digraphs with exactly two cycles.
Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for every \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament, all the intervals of which are trivial, is indecomposable; otherwise, it is decomposable. A critical tournament is an indecomposable tournament \(T\) of cardinality \(\geq 5\) such that for any vertex \(x\) of \(T\), the tournament \(T – x\) is decomposable. The critical tournaments are of odd cardinality and for all \(n \geq 2\) there are exactly three critical tournaments on \(2n + 1\) vertices denoted by \(T_{2n+1}\), \(U_{2n+1}\), and \(W_{2n+1}\). The tournaments \(T_5\), \(U_5\), and \(W_5\) are the unique indecomposable tournaments on 5 vertices. We say that a tournament \(T\) embeds into a tournament \(T’\) when \(T\) is isomorphic to a subtournament of \(T’\). A diamond is a tournament on 4 vertices admitting only one interval of cardinality 3. We prove the following theorem: if a diamond and \(T_5\) embed into an indecomposable tournament \(T\), then \(W_5\) and \(U_5\) embed into \(T’\). To conclude, we prove the following: given an indecomposable tournament \(T\) with \(|V(T)| \geq 7\), \(T\) is critical if and only if only one of the tournaments \(T_7\), \(U_7\), or \(W_7\) embeds into \(T\).
Let \(\lambda K_{m,n}\) be a complete bipartite multigraph with two partite sets having \(m\) and \(n\) vertices, respectively. A \(K_{p,q}\)-factorization of \(\lambda K_{m,n}\) is a set of edge-disjoint \(K_{p,q}\)-factors of \(\lambda K_{m,n}\) which is a partition of the set of edges of \(\lambda K_{m,n}\). When \(\lambda = 1\), Martin, in paper [Complete bipartite factorisations by complete bipartite graphs, Discrete Math., \(167/168 (1997), 461-480]\), gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will give similar necessary conditions for \(\lambda K_{m,n}\) to have a \(K_{p,q}\)-factorization, and prove the necessary conditions are always sufficient in many cases.
In this paper, we determine upper and lower bounds for the number of independent sets in a bicyclic graph in terms of its order. This
gives an upper bound for the total number of independent sets in a connected graph which contains at least two cycles. In each case, we characterize the extremal graphs.
Let \(G\) be a connected graph of order \(n\). Denote \(p_u(G)\) the order of a longest path starting at vertex \(u\) in \(G\). In this paper, we prove that if \(G\) has more than \(t\binom{k}{2} + \binom{p+1}{2} + (n-k-1)\) edges, where \(k \geq 2\), \(n = t(k-1) + p + 1\), \(t \geq 0\) and \(0 \leq p \leq k-1\), then \(p_u(G) > k\) for each vertex \(u\) in \(G\). By this result, we give an alternative proof of a result obtained by P. Wang et al. that if \(G\) is a 2-connected graph on \(n\) vertices and with more than \(t\binom{k-2}{2} + \binom{p}{2} + (2n – 3)\) edges, where \(k \geq 3\), \(n-2 = t(k-2) + p\), \(t \geq 0\) and \(0 \leq p \leq k-2\), then each edge of \(G\) lies on a cycle of order more than \(k\).
In this paper, we give some identities involving the harmonic numbers and the inverses of binomial coefficients.
In this paper, a new efficient computational algorithm is presented for solving cyclic heptadiagonal linear systems based on using the heptadiagonal linear solver and Sherman–Morrison–Woodbury formula. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Two numerical examples are presented for illustration.
Let \(G\) be a graph, and let \(a, b\), and \(k\) be nonnegative integers with \(0 \leq a \leq b\). A graph \(G\) is called an \((a, b, k)\)-critical graph if after deleting any \(k\) vertices of \(G\), the remaining graph of \(G\) has an \([a, b]\)-factor. In this paper, we prove that if \(\delta(G) \geq a + k\) and \(\alpha(G) \leq \frac{4b(\delta(G)-a+1-1)}{(a+1)^2}\), then \(G\) is an \((a, b, k)\)-critical graph. Furthermore, it is shown that the result in this paper is best possible in some sense.
A characterization of \(B\)-H-unretractive bipartite graphs is given. Based on this, it is proved that there is no bipartite graph with endotype \(1 \pmod{4}\).
In a graph \(G = (V, E)\), an independent set is a subset \(I\) of \(V(G)\) such that no two vertices in \(I\) are adjacent. A maximum independent set is an independent set of maximum size. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we study the problem of determining the largest and the second largest numbers of maximum independent sets among all quasi-tree graphs and quasi-forest graphs. Extremal graphs achieving these values are also given.
The notions of sum labelling and sum number of graphs were introduced by F. Harary [1] in 1990. A mapping \(f\) is called a sum labelling of a graph \(G(V, E)\) if it is an injection from \(V\) to a set of positive integers such that \(uv \in E\) if and only if there exists a vertex \(w \in V\) such that \(f(w) = f(x) + f(y)\). In this case, \(w\) is called a working vertex. If \(f\) is a sum labelling of \(G\) with \(r\) isolated vertices, for some nonnegative integer \(r\), and \(G\) contains no working vertex, \(f\) is defined as an exclusive sum labelling of the graph \(G\) by M. Miller et al. in paper [2]. The least possible number \(r\) of such isolated vertices is called the exclusive sum number of \(G\), denoted by \(\epsilon(G)\). If \(\epsilon(G) = \Delta(G)\), the labelling is called \(\Delta\)-optimum exclusive sum labelling and the graph is said to be \(\Delta\)-optimum summable, where \(\Delta = \Delta(G)\) denotes the maximum degree of vertices in \(G\). By using the notion of \(\Delta\)-optimum forbidden subgraph of a graph, the exclusive sum numbers of crown \(C_n \odot K_1\) and \((C_n \odot K_1)\) are given in this paper. Some \(\Delta\)-optimum forbidden subgraphs of trees are studied, and we prove that for any integer \(\Delta \geq 3\), there exist trees not \(\Delta\)-optimum summable. A nontrivial upper bound of the exclusive sum numbers of trees is also given in this paper.
In this paper we obtain the Fibonacci length of amalgamated free products having as factors dihedral groups.
In [11], Zhu, Li, and Deng introduced the definition of implicit degree of a vertex \(v\), denoted by \(\text{id}(v)\). In this paper, we consider implicit degrees and the hamiltonicity of graphs and obtain that:
If \(G\) is a \(2\)-connected graph of order \(n\) such that \(\text{id}(u) + \text{id}(v) \geq n – 1\) for each pair of vertices \(u\) and \(v\) at distance \(2\), then \(G\) is hamiltonian, with some exceptions.
Let \(C_k\) denote a cycle of length \(k\) and let \(S_k\) denote a star with \(k\) edges. For graphs \(F\), \(G\), and \(H\), a \((G, H)\)-multidecomposition of \(F\) is a partition of the edge set of \(F\) into copies of \(G\) and copies of \(H\) with at least one copy of \(G\) and at least one copy of \(H\). In this paper, necessary and sufficient conditions for the existence of the \((C_k, S_k)\)-multidecomposition of a complete bipartite graph are given.
This paper investigates the number of boundary cubic inner-forest maps and presents some formulae for such maps with the size (number of edges) and the valency of the root-face as two parameters. Further, by duality, some corresponding results for rooted outer-planar maps are obtained. It is also an answer to the open problem in \([15]\) and corrects the result on boundary cubic inner-tree maps in \([15]\).
The following two theorems are proved:
A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a cylinder so that the \(m\) rows go around the cylinder, with one square removed, with the exception of the following boards:
(a) \(n\) is even,
(b) \(m \in \{1,2\}\)
(c) \(m = 4\) and the removed square is in row 2 or 3;
(d) \(m \geq 5\), \(n = 1\), and the removed square is in row 2, 3, …, or \(m-1\).
A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a torus with one square removed except boards with \(m\) and \(n\) both even and \(1 \times 1\),\(1 \times 2\) and \(2 \times 1\) boards.
An independent set \(S\) of a connected graph \(G\) is called a \emph{frame} if \(G – S\) is connected. If \(|S| = k\), then \(S\) is called a \emph{k-frame}. We prove the following theorem.
Let \(k \geq 2\) be an integer, \(G\) be a connected graph with \(V(G) = \{v_1, v_2, \ldots, v_n\}\), and \(\deg_G(u)\) denote the degree of a vertex \(u\). Suppose that for every \(3\)-frame \(S = \{v_a, v_b, v_c\}\) such that \(1 \leq a \leq b \leq c \leq n\), \(\deg_G(v_c) \leq a\), \(\deg_G(v_b) \leq b-1\), and \(\deg_G(v_c) \leq c – 2\), it holds that\[\deg_G(v_a) + \deg_G(v_b) + \deg_G(v_c) – |N(v_a) \cap N(v_b) \cap N(v_c)| \geq |G| – k + 1.\] Then \(G\) has a spanning tree with at most \(k\)-leaves. Moreover, the condition is sharp.
This theorem is a generalization of the results of E. Flandrin, H.A. Jung, and H. Li (Discrete Math. \(90 (1991), 41-52)\) and of A. Kyaw (Australasian Journal of Combinatorics. \(37 (2007), 3-10)\) for traceability.
In this paper, the estimations of maximum genus orientable embeddings of graphs are studied, and an exponential lower bound for such numbers is found. Moreover, such two extremal embeddings (i.e., the maximum genus orientable embedding of the current graph and the minimum genus orientable embedding of the complete graph) are sometimes closely related to each other. As applications, we estimate the number of minimum genus orientable embeddings for the complete graph by estimating the number of maximum genus orientable embeddings for the current graph.
In this article, we characterize for which finite commutative rings \(R\), The zero-divisor graph \(\Gamma(R)\),The line graph \(L(\Gamma(R))\), The complement graph \(\overline{\Gamma(R)}\), and The line graph for the complement graph \(L(\overline{\Gamma(R)})\).
The energy of a graph \(G\) is defined as the sum of the absolute values of all the eigenvalues of the graph. In this paper, we consider the energy of the \(3\)-circulant graphs, and obtain a computation formula, and establish new results for a certain class of circulant graphs. At the same time, we give a conjecture: The largest energy of circulant graphs relates with their components.
In this paper, we present two criteria for a sequence lying along a ray of a combinatorial triangle to be unimodal, and give a correct
proof for the result of Belbachir and Szalay on unimodal rays of the generalized Pascal’s triangle.
In this paper, we introduce the notion of derivation in lattice implication algebra, and consider the properties of derivations in lattice implication algebras. We give an equivalent condition to be a derivation of a lattice implication algebra. Also, we characterize the fixed set \(Fix_d(L)\) and \(Kerd\) by derivations. Moreover, we prove that if \(d\) is a derivation of a lattice implication algebra, every filter \(F\) is \(d\)-invariant.
In this paper, we derive a family of identities on the arbitrary subscripted Fibonacci and Lucas numbers. Furthermore, we construct the tridiagonal and symmetric tridiagonal family of matrices whose determinants form any linear subsequence of the Fibonacci numbers and Lucas numbers. Thus, we give a generalization of the results presented in Nalli and Civciv [A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos, Solitons and Fractals \(2009;40(1):355 .61]\) and Cahill and Narayan [N. D. Cahill, D. A. Narayan, Fibonacci and Lucas numbers as tridiagonal matrix determinants, The Fibonacci Quarterly, \(2004;42(1):216–221]\).
A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) \(G\) with vertex set \(V(G)\) is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex \(x \in V(G)\) such that \(G – x\) is a tree (respectively, forest). In this paper, we determine the second largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.
For a poset \(P = (X, \leq_ P)\), the strict-double-bound graph (\(sDB\)-graph \(sDB(P)\)) is the graph on \(X\) for which vertices \(u\) and \(v\) of \(sDB(P)\) are adjacent if and only if \(u \neq v\) and there exist \(x\) and \(y\) in \(X\) distinct from \(u\) and \(v\) such that \(x \leq_ P y\) and \(x \leq_P v \leq_P y\). The strict-double-bound number \(\zeta(G)\) of a graph \(G\) is defined as \(\min\{n; G \cup \overline{K}_n \text{ is a strict-double-bound graph}\}\).
We obtain that for a spider \(S_{n,m}\) (\(n,m > 3\)) and a ladder \(L_n\) (\(n \geq 4\)), \(\left\lceil2\sqrt{nm}\right\rceil \leq \zeta(S_{n,m}) \leq n+m\), \(\zeta(S_{n,n}) = 2n\), and \(\left\lceil 2\sqrt{3n+2}\right\rceil \leq \zeta(L_n) \leq 2n\).
We conjectured in \([3]\) that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the \(3\)-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete \(k\)-partite, and matrogenic cyclic graphs are weakly split.
Let \((X,\mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition and let \(G_i\), \(i = 1,\ldots,\mu\), be nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i | B \in \mathcal{B}\}\), where \(\mathcal{B_i}\) is a subgraph of \(B\) isomorphic to \(G_i\). A \(\{G_1,G_2,\ldots,G_\mu\}\)-metamorphosis of \((X,\mathcal{B})\) is a rearrangement, for each \(i=1,\ldots,\mu\), of the edges of \(\bigcup_{B\in B}(E(B)\setminus\mathcal{B}_i))\) into a family \(\mathcal{F}_i\) of copies of \(G_i\) with a leave \(L_i\), such that \((X,\mathcal{B}_i \cup \mathcal{F}_i,L_i)\) is a maximum packing of \(\lambda H\) with copies of \(G_i\). In this paper, we give a complete answer to the existence problem of an \(S_\lambda(2,4,7)\) having a \(\{C_4, K_3 + e\}\)-metamorphosis.
For a positive integer \(m\), where \(1 \leq m \leq n\), the \(m\)-competition index (generalized competition index) of a primitive digraph \(D\) of order \(n\) is the smallest positive integer \(k\) such that for every pair of vertices \(x\) and \(y\), there exist \(m\) distinct vertices \(v_1, v_2, \ldots, v_m\) such that there exist walks of length \(k\) from \(x\) to \(v_i\) and from \(y\) to \(v_i\) for \(1 \leq i \leq m\). In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.
We give a new combinatorial bijection between a certain set of balanced modular tableaux of Gusein-Zade, Luengo, and Melle-Hernandez and \(k\)-ribbon shapes. In addition, we also use the Schensted algorithm for the rim hook tableaux of Stanton and White to write down an explicit generating function for these balanced modular tableaux.
A \((k;g)\)-cage is a graph with the minimum order among all \(k\)-regular graphs with girth \(g\). As a special family of graphs, \((k;g)\)-cages have a number of interesting properties. In this paper, we investigate various properties of cages, e.g., connectivity, the density of shortest cycles, bricks and braces.
Let \(G = (V, E)\) be a digraph with \(n\) vertices and \(m\) arcs without loops and multiarcs, \(V = \{v_1, v_2, \ldots, v_n\}\). Denote the outdegree and average \(2\)-outdegree of the vertex \(v_i\) by \(d^+_i\) and \(m^+_i\), respectively. Let \(A(G)\) be the adjacency matrix and \(D(G) = \text{diag}(d^+_1, d^+_2, \ldots, d^+_n)\) be the diagonal matrix with outdegrees of the vertices of the digraph \(G\). Then we call \(Q(G) = D(G) + A(G)\) the signless Laplacian matrix of \(G\). In this paper, we obtain some upper and lower bounds for the spectral radius of \(Q(G)\), which is called the signless Laplacian spectral radius of \(G\). We also show that some bounds involving outdegrees and the average \(2\)-outdegrees of the vertices of \(G\) can be obtained from our bounds.
Lee and Kong conjecture that if \(n \geq 1\) is an odd number, then \(St(a_1, a_0, \ldots, a_n)\) would be super edge-magic, and meanwhile they proved that the following graphs are super edge-magic: \(St(m,n)\) (\(n = 0 \mod (m+1)\)), \(St(1,k,n)\) (\(k = 1,2\) or \(n\)), \(St(2, k,n)\) (\(k = 2,3\)), \(St(1,1,k,n)\) (\(k = 2,3\)), \(St(k,2,2,n)\) (\(k = 1,2\)). In this paper, the conjecture is further discussed and it is proved that \(St(1,m,n)\), \(St(3,m,m+1)\), \(St(n,n+1,n+2)\) are super edge-magic, and under some conditions \(St(a_1, a_2, \ldots, a_{2n+1})\); \(St(a_1, a_2, \ldots, a_{4n+1})\), \(St(a_1, a_2, \ldots, a_{4n+3})\) are also super edge-magic.
We determine all connected odd graceful graphs of order \(\leq 6\). We show that if \(G\) is an odd graceful graph, then \(G \cup K_{m,n}\) is odd graceful for all \(m, n \geq 1\). We give an analogous statement to the graceful graphs statement, and we show that some families of graphs are odd graceful.
In this paper, we provide a method to obtain the lower bound on the number of distinct maximum genus embeddings of the complete bipartite graph \(K_{n,n}\) (\(n\) is an odd number), which, in some sense, improves the results of S. Stahl and H. Ren.
For positive integer \(n\), let \(f_3(n)\) be the least upper bound of the sums of the lengths of the sides of \(n\) cubes packed into a unit cube \(C\) in three dimensions in such a way that the smaller cubes have sides parallel to those of \(C\). In this paper, we improve the lower bound of \(f_3(n)\).
The transformation graph \(G^{+- -}\) of a graph \(G\) is the graph with vertex set \(V(G) \cup E(G)\), in which two vertices \(u\) and \(uv\) are joined by an edge if one of the following conditions holds: (i) \(u,v \in V(G)\) and they are adjacent in \(G\), (ii) \(u,v \in E(G)\) and they are not adjacent in \(G\), (iii) one of \(u\) and \(wv\) is in \(V(G)\) while the other is in \(E(G)\), and they are not incident in \(G\). In this paper, for any graph \(G\), we determine the independence number and the connectivity of \(G^{+- -}\). Furthermore, we show that for a graph \(G\) with no isolated vertices, \(G^{+- -}\) is hamiltonian if and only if \(G\) is not a star and \(G \not\in \{2K_2, K_2\}\).
We introduce quasi-almostmedian graphs as a natural nonbipartite generalization of almostmedian graphs. They are filling a gap between quasi-median graphs and quasi-semimedian graphs. We generalize some results of almostmedian graphs and deduce some results from a bigger class of quasi-semimedian graphs. The consequence of this is another characterization of almostmedian graphs as well as two new characterizations of quasi-median graphs.
In this note, we establish a convolution formula for Bernoulli polynomials in a new and brief way, and some known results are derived as a special case.
In this study, we define the generalized \(k\)-order Fibonacci matrix and the \(n \times n\) generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. We find the inverse of the generalized Pascal matrix \(\mathcal{F}_n(GF)\) associated with generalized \(\mathcal{F}\)-nomial coefficients. In the last section, we factorize this matrix via the generalized \(k\)-order Fibonacci matrix and give illustrative examples for these factorizations.
The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let \(\mathcal{G}\) be the set of unicyclic graphs of order \(n\) with girth \(g\). For all integers \(n\) and \(g\) with \(5 \leq g \leq n – 6\), we determine the first \(|\frac{g}{2}| + 3\) spectral radii of unicyclic graphs in the set \(\mathcal{U}_n^g\).
In this paper, we consider labelings of graphs in which the label on an edge is the absolute value of the difference of its vertex labels. Such a labeling using \(\{0,1,2,\ldots,k-1\}\) is called \(k\)-equitable if the number of vertices (resp. edges) labeled \(i\) and the number of vertices (resp. edges) labeled \(j\) differ by at most one and is called \(k\)-balanced if the number of vertices labeled \(i\) and the number of edges labeled \(j\) differ by at most one. We determine which graphs in certain families are \(k\)-equitable or \(k\)-balanced and we give also some necessary conditions on these two labelings.
The study of chromatically unique graphs has been drawing much attention and many results are surveyed in \([4, 12, 13]\). The notion of adjoint polynomials of graphs was first introduced and applied to the study of the chromaticity of the complements of the graphs by Liu \([17]\) (see also \([4]\)). Two invariants for adjoint equivalent graphs that have been employed successfully to determine chromatic unique graphs were introduced by Liu \([17]\) and Dong et al. \([4]\) respectively. In the paper, we shall utilize, among other things, these two invariants to investigate the chromaticity of the complement of the tadpole graphs \(C_n(P_m)\), the graph obtained from a path \(P_m\) and a cycle \(C_n\) by identifying a pendant vertex of the path with a vertex of the cycle. Let \(\bar{G}\) stand for the complement of a graph \(G\). We prove the following results:
1. The graph \(\overline{{{C}_{n-1}(P_2)}}\) is chromatically unique if and only if \(n \neq 5, 7\).
2. Almost every \(\overline{{C_n(P_m)}}\) is not chromatically unique, where \(n \geq 4\) and \(m \geq 2\).
An \(L(2,1)\)-labelling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x,y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x,y) = 2\), where \(d(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \((2,1)\)-labelling number \(\lambda(G)\) of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2,1)\)-labelling with \(\max\{f(v) : v \in V(G)\} = k\). Griggs and Yeh conjecture that \(\lambda(G) \leq \Delta^2\) for any simple graph with maximum degree \(\Delta \geq 2\). This article considers the graphs formed by the cartesian product of \(n\) (\(n \geq 2\) graphs. The new graph satisfies the above conjecture (with minor exceptions). Moreover, we generalize our results in [19].
In this study, we first define new sequences named \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas sequences. After that, by using these sequences, we establish \((s, t)\)-Jacobsthal and \((s, t)\) Jacobsthal-Lucas matrix sequences. Finally, we present some important relationships between these matrix sequences.
Several transformations about \(_\gamma F_6(1)\)-series are established by applying the modified Abel lemma on summation by parts. As a consequence, a reciprocal relation on balanced \(_3F_2(1)\)-series is derived, which may also be considered as a nonterminating extension of Saalschütz’s theorem (1891).
Let \( G \) be the one-point union of two cycles and suppose \( G \) has \( n \) edges. We show via various graph labelings that there exists a cyclic \( G \)-decomposition of \( K_{2nt+1} \) for every positive integer \( t \).
Recently Ozbal and Firat [22] introduced the notion of symmetric \( f \) bi-derivation of a lattice. They give illustrative examples and they also characterized the distributive lattice by symmetric \( f \) bi-derivation. In this paper, we define the isotone symmetric \( f \) bi-derivation and obtain some interesting results about isotoneness. We also provide the relations between distributive, modular, and isotone lattices through symmetric \( f \) bi-derivation.
In 2003, Lee, Wang and Wen found a non-edge-magic simple connected cubic graph which satisfying the necessary condition of edge-magicness by using computer search. They asked for a mathematical proof. In this paper, we will provide such a proof.
Let \( G \) be a graph and let \( f \) be a positive integer-valued function defined on \( V(G) \) such that \( 1 \leq a \leq f(x) \leq b \leq 2a \) for every \( x \in V(G) \). If \( t(G) \geq \frac{b^2}{a} \), \( |V(G)| \geq \frac{b^2}{a} + 1 \), and \( f(V(G)) \) is even, then \( G \) has an \( f \)-factor.
A general construction for \( t \)-SB(\(2t-1\), \(2t-2\)) designs is given. In addition, large sets of \( t \)-SB(\(v\), \(k\)) are discussed and some examples are provided.
For a poset \( P = (X, \leq_P) \), the strict-double-bound graph (\(sDB\)-graph) of \( P = (X, \leq_P) \) is the graph \( sDB(P) \) on \( X \) for which vertices \( u \) and \( v \) of \( sDB(P) \) are adjacent if and only if \( u \neq v \) and there exist \( x \) and \( y \) in \( X \) distinct from \( u \) and \( v \) such that \( x \leq u \leq y \) and \( x \leq v \leq y \). The strict-double-bound number \( \zeta(G) \) is defined as
\[
\zeta(G) = \min \{ n \mid G \cup N_n \text{ is a strict-double-bound graph} \},
\]
where \( N_n \) is the graph with \( n \) vertices and no edges.
In this paper we deal with strict-double-bound numbers of some graphs. For example, we obtain that
\[
\zeta(P_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 2 \text{)},
\]
\[
\zeta(C_n) = \lceil 2\sqrt{n} \rceil \text{ (} n \geq 4 \text{)},
\]
\[
\zeta(W_n) = \lceil 2\sqrt{n-1} \rceil \text{ (} n \geq 5 \text{)},
\]
and
\[
\zeta(G + K_n) = \zeta(G)
\]
for a graph \( G \) with no isolated vertices.
The metric dimension of a graph \(G\), denoted by \(\text{dim}(G)\), is the minimum number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. For a graph \(G\) and its complement \(\overline{G}\), each of order \(n \geq 4\) and connected, we show that
\[
2 \leq \text{dim}(G) + \text{dim}(\overline{G}) \leq 2(n-3).
\]
It is readily seen that \(\text{dim}(G) + \text{dim}(\overline{G}) = 2\) if and only if \(n = 4\). We characterize graphs satisfying
\[
\text{dim}(G) + \text{dim}(\overline{G}) = 2(n-3)
\]
when \(G\) is a tree or a unicyclic graph.
Whist tournament designs are known to exist for all \( v \equiv 0,1 \pmod{4} \). Much less is known about the existence of \(\mathbb{Z}\)-cyclic whist designs. Previous studies \([5, 6]\) have reported on all \(\mathbb{Z}\)-cyclic whist designs for \( v \in \{4,5,8,9,12,13,16,17,20,21,24,25\} \). This paper is a report on all \(\mathbb{Z}\)-cyclic whist tournament designs on 28 players, including a detailed summary of all known whist specializations related to a 28 player \(\mathbb{Z}\)-cyclic whist design. Our study shows that there are \( 7,910,127 \) \(\mathbb{Z}\)-cyclic whist designs on 28 players. Of these designs, \( 2,568,510 \) possess the Three Person Property, \( 240,948 \) possess the Triplewhist Property and none possess the Balancedwhist Property. Introduced here is the concept of the mirror image of a \(\mathbb{Z}\)-cyclic whist design. In general, utilization of this concept reduces the computer search for \(\mathbb{Z}\)-cyclic whist designs by nearly fifty percent.
Let \( S \) be an orthogonal polygon in the plane bounded by a simple closed curve. Assume that every two boundary points of \( S \) have a common staircase illuminator whose edges are north and east. Then \( S \) contains a staircase path \( \mu_0 \) whose edges are north and east such that \( \mu_0 \) illumines every point of \( S \). Without the requirement that the illuminators share a common direction, the result fails.
The upper domination Ramsey number \(u(3, 3, 3)\) is the smallest integer \(n\) such that every \(3\)-coloring of the edges of complete graph \(K_n\) contains a monochromatic graph \(G\) with \(\Gamma(\overline{G}) \geq 3\), where \(\Gamma(\overline{G})\) is the maximum order over all the minimal dominating sets of the complement of \(G\). In this note, with the help of computers, we determine that \(U(3, 3, 3) = 13\), which improves the results that \(13 \leq U(3, 3, 3) \leq 14\) provided by Michael A. Henning and Ortrud R. Oellermann.
Let \(G\) be a graph of order \(n \geq 4k+8\), where \(k\) is a positive integer with \(kn\) even and \(\delta(G) > k+1\). We show that if \(max\{d_G(u),d_G(v)\} > {n}/{2}\) for each pair of nonadjacent vertices \(u,v\), then \(G\) has a connected \([k, k+1]\)-factor excluding any given edge \(e\).
A \( k \)-edge labeling of a graph \( G \) is a function \( f \) from the edge set \( E(G) \) to the set of integers \( \{0, \ldots, k-1\} \). Such a labeling induces a labeling \( f \) on the vertex set \( V(G) \) by defining \( f(v) = \sum f(e) \), where the summation is taken over all the edges incident on the vertex \( v \) and the value is reduced modulo \( k \). Cahit calls this labeling edge-\( k \)-equitable if \( f \) assigns the labels \( \{0, \ldots, k-1\} \) equitably to the vertices as well as edges.
If \( G_1, \ldots, G_T \) is a family of graphs having a graph \( H \) as an induced subgraph, then by \( H \)-union \( G \) of this family we mean the graph obtained by identifying all the corresponding vertices as well as edges of the copies of \( H \) in \( G_1, \ldots, G_T \).
In this paper we prove that the \( \overline{K}_n \)-union of gears is edge-\( 3 \)-equitable.
Let \( k \) be a positive integer and let \( G \) be a simple graph with vertex set \( V(G) \). If \( v \) is a vertex of \( G \), then the open \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}(v) \), is the set \( N_{k,G}(v) = \{u \mid u \neq v \text{ and } d(u, v) \leq k\} \). The closed \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}[v] \), is \( N_{k,G}[v] = N_{k,G}(v) \cup \{v\} \). A function \( f: V(G) \to \{-1,1\} \) is called a \({signed\; distance \; k -dominating\; function}\) if \( \sum_{u \in N_{k,G}(v)} f(u) \geq 1 \) for each vertex \( v \in V(G) \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed distance \( k \)-dominating functions on \( G \) with the property that \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V(G) \) is called a \({signed\; distance \; k -dominating \;family}\) (of functions) on \( G \). The maximum number of functions in a signed distance \( k \)-dominating family on \( G \) is the \({signed\; distance \; k -domatic\; number}\) of \( G \), denoted by \( d_{k,s}(G) \). Note that \( d_{1,s}(G) \) is the classical signed domatic number \( d_s(D) \). In this paper, we initiate the study of signed distance \( k \)-domatic numbers in graphs and we present some sharp upper bounds for \( d_{k,s}(G) \).
We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacency matrix and automorphism group.
In this paper, we consider a variation of toughness, and prove stronger results for the existence of \([a, b]\)-factors. Furthermore, we show that the results are sharp in some sense.
A decomposition \( \mathcal{D} \) of a graph \( H \) by a graph \( G \) is a partition of the edge set of \( H \) such that the subgraph induced by the edges in each part of the partition is isomorphic to \( G \). The intersection graph \( I(\mathcal{D}) \) of the decomposition \( \mathcal{D} \) has a vertex for each part of the partition and two parts \( A \) and \( B \) are adjacent if and only if they share a common node in \( H \). If \( I(\mathcal{D}) \cong H \), then \( \mathcal{D} \) is an automorphic decomposition of \( H \). If \( n(G) = \chi(H) \) as well, then we say that \( \mathcal{D} \) is a fully automorphic decomposition. In this paper, we examine the question of whether a fully automorphic host will have an even degree of regularity. We also give several examples of fully automorphic decompositions as well as necessary conditions for their existence.
A modular \(k\)-coloring, \(k \geq 2\), of a graph \(G\) without isolated vertices is a coloring of the vertices of \(G\) with the elements in \(\mathbb{Z}_k\) (where adjacent vertices may be colored the same) having the property that for every two adjacent vertices in \(G\) the sums of the colors of their neighbors are different in \(\mathbb{Z}_k\). The minimum \(k\) for which \(G\) has a modular \(k\)-coloring is the modular chromatic number mc\((G)\) of \(G\). It is known that \(2 \leq \text{mc}(T) \leq 3\) for every nontrivial tree \(T\). We present an efficient algorithm that computes the modular chromatic number of a given tree.
In this note, we consider the \(i\)-block intersection graphs (\(i\)-BIG) of a universal friendship \(3\)-hypergraph and show that they are pancyclic for \(i = 1,2\). We also show that the \(1\)-BIG of a universal friendship \(3\)-hypergraph is Hamiltonian-connected.
We study samples \(\Gamma = (\Gamma_1, \ldots, \Gamma_n)\) of length \(n\) where the letters \(\Gamma_i\) are independently generated according to the geometric distribution \(\mathbb{P}(\Gamma_j = i) = pq^{i-1}\), for \(1 \leq j \leq n\), with \(p+q=1\) and \(0<p<1\). An \({up-smooth\; sample}\) \(\Gamma\) is a sample such that \(\Gamma_{i+1}- \Gamma_i \leq 1\). We find generating functions for the probability that a sample of \(n\) geometric variables is up-smooth, with or without a specified first letter. We also extend the up-smooth results to words over an alphabet of \(k\) letters and to compositions of integers. In addition, we study smooth samples \(T\) of geometric random variables, where the condition now is \(|\Gamma_{i+1}- \Gamma_i| \leq 1\).
A Roman dominating function on a graph \(G = (V, E)\) is a function \(f : V \to \{0,1,2\}\) satisfying the condition that every vertex \(u \in V\) for which \(f(u) = 0\) is adjacent to a vertex \(v\) for which \(f(v) = 2\). The weight of a Roman dominating function is the value \(f(V) = \sum_{u \in V} f(u)\). The Roman domination number, \(\gamma_R(G)\), of \(G\) is the minimum weight of a Roman dominating function on \(G\). In this paper, we study those graphs for which the removal of any pair of vertices decreases the Roman domination number. A graph \(G\) is said to be \emph{Roman domination bicritical} or just \(\gamma_R\)-bicritical, if \(\gamma_R(G – \{v,u\}) < \gamma_R(G)\) for any pair of vertices \(v,u \in V\). We study properties of \(\gamma_R\)-bicritical graphs, and we characterize \(\gamma_R\)-bicritical trees and unicyclic graphs.
The van der Waerden number \(W(r, k)\) is the least integer \(N\) such that every \(r\)-coloring of \(\{1, 2, \ldots, N\}\) contains a monochromatic arithmetic progression of length at least \(k\). Rabung gave a method to obtain lower bounds on \(W(2, k)\) based on quadratic residues, and performed computations on all primes no greater than \(20117\). By improving the efficiency of the algorithm of Rabung, we perform the computation for all primes up to \(6 \times 10^7\), and obtain lower bounds on \(W(2, k)\) for \(k\) between \(11\) and \(23\).
Let \( G = (V, E) \) be a graph with chromatic number \( k \). A dominating set \( D \) of \( G \) is called a chromatic transversal dominating set (ctd-set) if \( D \) intersects every color class of any \( k \)-coloring of \( G \). The minimum cardinality of a ctd-set of \( G \) is called the chromatic transversal domination number of \( G \) and is denoted by \( \gamma_{ct}(G) \). In this paper, we obtain sharp upper and lower bounds for \( \gamma_{ct} \) for the Mycielskian \( \mu(G) \) and the shadow graph \( \text{Sh}(G) \) of any graph \( G \). We also prove that for any \( c \geq 2 \), the decision problem corresponding to \( \gamma_{ct} \) is NP-hard for graphs with \( \chi(G) = c \).
Let \( G(V, E) \) be a simple graph, and let \( f \) be an integer function defined on \( V \) with \( 1 \leq f(v) \leq d(v) \) for each vertex \( v \in V \). An \( f \)-edge covered colouring is an edge colouring \( C \) such that each colour appears at each vertex \( v \) at least \( f(v) \) times. The maximum number of colours needed to \( f \)-edge covered colour \( G \) is called the \( f \)-edge covered chromatic index of \( G \) and denoted by \( \chi_{fc}'(G) \). Any simple graph \( G \) has an \( f \)-edge covered chromatic index equal to \( \delta_f \) or \( \delta_f – 1 \), where \( \delta_f = \min \left\{\left\lfloor\frac{d(v)}{f(v)}\right\rfloor : v \in V(G)\right\} \). Let \( G \) be a connected and not complete graph with \( \chi_{fc}’ = \delta_f – 1 \). If for each \( u, v \in V \) and \( e = uv \notin E \), we have \( \chi_{fc}'(G+e) > \chi_{fc}'(G) \); then \( G \) is called an \( f \)-edge covered critical graph. In this paper, some properties of \( f \)-edge covered critical graphs are discussed. It is proved that if \( G \) is an \( f \)-edge covered critical graph, then for each \( u, v \in V \) and \( e = uv \notin E \) there exists \( w \in \{u, v\} \) with \( d(w) \leq \delta_f(f(w) + 1) – 2 \) such that \( w \) is adjacent to at least \( \max \left\{d(w) – \delta_f f(w) + 1, (f(w) + 2)d(w) – \delta_f(f(w) + 1)^2 + f(w) + 3\right\} \) vertices which are all \( \delta_f \)-vertices in \( G \).
An almost-bipartite graph is a non-bipartite graph with the property that the removal of a particular single edge renders the graph bipartite. A graph labeling of an almost-bipartite graph \(G\) with \(n\) edges that yields cyclic \(G\)-decompositions of the complete graph \(K_{2nt+1}\) was recently introduced by Blinco, El-Zanati, and Vanden Eynden. They called such a labeling a \(\gamma\)-labeling. Here we show that the class of almost-bipartite graphs obtained from a path with at least \(3\) edges by adding an edge joining distinct vertices of the path an even distance apart has a \(\gamma\)-labeling.
The locally twisted cube \(LTQ_n\) is an important variation of hypercube and possesses many desirable properties for interconnection networks. In this paper, we investigate the problem of embedding paths in faulty locally twisted cubes. We prove that a path of length \(l\) can be embedded between any two distinct vertices in \(LTQ_n – F\) for any faulty set \(F \subseteq V(LTQ_n) \cup E(LTQ_n)\) with \(|F| \leq n-3\) and any integer \(l\) with \(2^{n-1} \leq l \leq |V(LTQ_n – F)| – 1\) for any integer \(n > 3\). The result is tight with respect to the two bounds on path length \(l\) and faulty set size \(|F|\) for a successful embedding.
A \(G\)-design is a partition of \(E(K_v)\) in which each element induces a copy of \(G\). The existence of \(G\)-designs with the additional property that they contain no proper subsystems has been previously settled when \(G \in \{K_3, K_4 – e\}\). In this paper, the existence of \(P_m\)-designs which contain no proper subsystems is completely settled for every value of \(m\) and \(v\).
The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\). In this paper, we give a sharp lower bound on the Randić index of cacti with perfect matchings.
Let \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) be the \((2v+1)\)-dimensional affine-singular symplectic space over the finite field \(\mathbb{F}_q\) and let \(\text{ASp}_{2v+1}(\mathbb{F}_q)\) be the affine-singular symplectic group of degree \(2v+1\) over \(\mathcal{F}_q\). For any orbit \(O\) of flats under \(\text{ASp}_{2v+1}(\mathbb{F}_q)\), let \(\mathcal{L}\) be the set of all flats which are intersections of flats in \(O\) such that \(O \subseteq \mathcal{L}\) and assume the intersection of the empty set of flats in \(\text{ASG}(2v+1,v;\mathbb{F}_q)\) is \(\mathbb{F}_q^{2v+1}\). By ordering \(\mathcal{L}\) by ordinary or reverse inclusion, two lattices are obtained. This article discusses the relations between different lattices, classifies their geometricity, and computes their characteristic polynomial.
Let \(\gamma_c(G)\) be the connected domination number of \(G\).A graph is \(k\)-\(\gamma_c\)-critical if \(\gamma_c(G) = k\) and \(\gamma_c(G + uv) < \gamma_c(G)\) for any nonadjacent pair of vertices \(u\) and \(v\) in the graph \(G\). In this paper, we show that the diameter of a \(k\)-\(\gamma_c\)-critical graph is at most \(k\) and this upper bound is sharp.
A \(b\)-coloring of a graph \(G\) by \(k\) colors is a proper \(k\)-coloring of the vertices of \(G\) such that in each color class there exists a vertex having neighbors in all the other \(k-1\) color classes. The \(b\)-chromatic number \(\varphi(G)\) of a graph \(G\) is the maximum \(k\) for which \(G\) has a \(b\)-coloring by \(k\) colors. This concept was introduced by R.W. Irving and D.F. Manlove in \(1999\). In this paper, we study the \(b\)-chromatic numbers of the cartesian products of paths and cycles with complete graphs and the cartesian product of two complete graphs.
Let \(K_{d,d}\) be a complete bipartite digraph. In this paper, we determine the exact value of the domination number in iterated line digraph of \(K_{d,d}\).
A total coloring of a simple graph \(G\) is called adjacent vertex distinguishing if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) differs from the set of colors assigned to the vertices and the edges incident to \(v\). In this paper, we shall prove that the adjacent vertex distinguishing total chromatic number of an outer plane graph with \(\Delta \leq 5\) is \(\Delta+2\) if \(G\) has two adjacent maximum degree vertices, otherwise it is \(\Delta+1\).
Let \(P_j(n)\) denote the number of representations of \(n\) as a sum of \(j\) pentagonal numbers. We obtain formulas for \(P_j(n)\) when \(j = 2\) and \(j = 3\).
Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. We study some variations of this concept in which the configuration of guards induce total dominating sets. We consider two models of the problem: one in which only one guard moves at a time and one in which all guards may move simultaneously. A number of upper and lower bounds are given for the number of guards required.
Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\) then the subgraph is called a spanning subgraph of \(G\). A spanning subgraph \(H\) of \(G\) is called an \(F\)-factor if each component of \(H\) is isomorphic to \(F\). Further, if there exists a subgraph of \(G\) whose vertex set is \(V(G)\) and can be partitioned into \(F\)-factors, then it is called a \(\lambda\)-fold \(F\)-factor of \(G\), denoted by \(S_\lambda(1,F,G)\). A large set of \(\lambda\)-fold \(F\)-factors of \(G\), denoted by \(LS_\lambda(1,F,G)\), is a partition \(\{\mathcal{B}_i\}_i\) of all subgraphs of \(G\) isomorphic to \(F\), such that each \((X,\mathcal{B}_i)\) forms a \(\lambda\)-fold \(F\)-factor of \(G\). In this paper, we investigate \(LS_\lambda(1,K_{1,3},K_{v,v})\) for any index \(\lambda\) and obtain existence results for the cases \(v = 4t, 2t + 1, 12t+6\) and \(v \geq 3\).
In this paper, we give some interesting identities on the Bernoulli and the Euler numbers and polynomials by using reflection symmetric properties of Euler and Bernoulli polynomials. To derive our identities, we investigate some properties of the fermionic \(p\)-adic integrals on \(\mathbb{Z}_p\).
For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:
\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]
A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).
We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in [11] if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.
In \(1989\), Zhu, Li, and Deng introduced the definition of implicit degree, denoted by \(\text{id}(v)\), of a vertex \(v\) in a graph \(G\) and they obtained sufficient conditions for a graph to be hamiltonian with the implicit degrees. In this paper, we prove that if \(G\) is a \(2\)-connected graph of order \(n\) with \(\alpha(G) \leq n/2\) such that \(\text{id}(v) \geq (n-1)/2\) for each vertex \(v\) of \(G\), then \(G\) is hamiltonian with some exceptions.
The compact, Fredholm, and isometric weighted composition operators are characterized in this paper.
We discuss the chromaticity of one family of \(K_4\)-homeomorphs with exactly two non-adjacent paths of length two, where the other four paths are of length greater than or equal to three. We also give a sufficient and necessary condition for the graphs in the family to be chromatically unique.
In this paper, we deduced the following new Stirling series:
\[ n! \sim \sqrt{2n\pi} (\frac{n}{2})^n exp(\frac{1}{12n+1}[1 + \frac{1}{12n} (1+\frac{\frac{2}{5}}{n} + \frac{\frac{29}{150}}{n^2} – \frac{\frac{62}{2625}}{n^3} – \frac{\frac{9173}{157500}}{n^4} +\ldots )^{-1}]) ,\]
which is faster than the classical Stirling’s series.
For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:
\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]
A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).
We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in \([11]\) if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.
The general neighbor-distinguishing total chromatic number \(\chi”_{gnd}(G)\) of a graph \(G\) is the smallest integer \(k\) such that the vertices and edges of \(G\) can be colored by \(k\) colors so that no adjacent vertices have the same set of colors. It is proved in this note that \(\chi”_{gnd}(G) = \lceil \log_2 \chi(G) \rceil + 1\), where \(\chi(G)\) is the vertex chromatic number of \(G\).
A sequence \(A\) is a \(B_h^*[g]\) sequence if the coefficients of \((\sum_{a\in A}(z)^a)^h\) are bounded by \(g\). The standard Sidon sequence is a \(B[2]\) sequence. Finite Sidon sequences are called Golomb rulers, which are found to have many applications such as error correcting codes, radio frequency selection, and radio antennae placement. Let \(R_h(g,n)\) be the largest cardinality of a \(B[g]\) sequence contained in \(\{1,2,\ldots,n\}\), and \(F(h,g,k) = \min\{n : R_h(g,n) \geq k\}\). In this paper, computational techniques are applied to construct optimal generalized Sidon sequences, and \( 49\) new exact values of \(F(2,g,k)\) are found.
Recently, Chu \([5]\) derived two families of terminating \(_2F_1(2)\)-series identities. Their \(q\)-analogues will be established in this paper.
Let \(H\), \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \(G-GD_\lambda(H)\), is a partition of all the edges of \(H\) into subgraphs (called \(G\)-blocks), each of which is isomorphic to \(G\). A large set of \(G-GD_\lambda(H)\), denoted by \(G-LGD_\lambda(H)\), is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \(G-GD_\lambda(H)\)s. In this paper, we determine the existence spectrums for \(K_{2,2}-LGD_\lambda(K_{m,n})\).
A binary vertex coloring (labeling) \(f: V(G) \to \mathbb{Z}_2\) of a graph \(G\) is said to be friendly if the number of vertices labeled 0 is almost the same as the number of vertices labeled 1. This friendly labeling induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(uv) = f(u)f(v)\) for all \(uv \in E(G)\). Let \(e_f(i) = |\{uv \in E(G) : f^*(uv) = i\}|\) be the number of edges of \(G\) that are labeled \(i\). The product-cordial index of the labeling \(f\) is the number \(pc(f) = |e_f(0) – e_f(1)|\). The product-cordial set of the graph \(G\), denoted by \(PC(G)\), is defined by
\[PC(G) = \{pc(f): f \text{ is a friendly labeling of } G\}.\]
In this paper, we will determine the product-cordial sets of long grids \(P_m \times P_n\), introduce a class of fully product-cordial trees and suggest new research directions in this topic.
In this paper, we investigate some interesting identities on the Euler numbers and polynomials arising from their generating functions and difference operators. Finally, we give some properties of Bernoulli and Euler polynomials by using \(p\)-adic integral on \(\mathbb{Z}_p\).
Let \(\pi\) be a finite projective plane of order \(n\). Consider the substructure \(\pi_{n+2}\) obtained from \(\pi\) by removing \(n+2\) lines (including all points on them) no three of which are concurrent. In this paper, firstly, it is shown that \(\pi_{n+2}\) is a B-L plane and it is also homogeneous. Let \(PG(3,2)\) be a finite projective \(3\)-space of order \(n\). The substructure obtained from \(PG(3,2)\) by removing a tetrahedron that is four planes of \(PG(3,n)\) no three of which are collinear is a finite hyperbolic \(3\)-space (Olgun-Ozgir [10]). Finally, we prove that any two hyperbolic planes with the same parameters are isomorphic in this hyperbolic \(3\)-space. These results appeared in the second author’s MSc thesis.
Let \(G\) be a graph of order \(n\), and let \(a\) and \(b\) be integers such that \(1 \leq a < b\). Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) < f(x) \leq b\) for each \(x \in V(G)\). Then \(G\) has a \((g, f)\)-factor if the minimum degree \(\delta(G) \geq \frac{(b-1)^2-(a+1)(a+b-1)}{a+1}\) ,\(n>\frac{(a+b)(a+b-1)}{a+1}\) and \(\max\{d_G(x), d_G(y)\} \geq \frac{(b-1)n}{a+b}\) for any two nonadjacent vertices \(x\) and \(y\) in \(G\). Furthermore, it is shown that the result in this paper is best possible in some sense.
In this note, we consider the on-line Ramsey numbers \(\overline{R}(P_n, P_m)\) for paths. Using a high-performance computing cluster, we calculated the values for off-diagonal numbers for paths of lengths at most \(8\). Also, we were able to check that \(\overline{R}(P_9, P_9) = 17\), thus solving the problem raised in [5].
In this paper, we determine the images of hyperbolic ellipses under the Möbius and harmonic Möbius transformations.
Given a disjoint union of some complete graphs, one can define a graph by choosing one vertex from each complete graph and making all of these vertices adjacent. This observation leads us to define a new operation on certain graphs. We compute the characteristic polynomial of the resulting graphs and indicate a method for computing the determinant of this matrix for obtaining the characteristic polynomial of new graphs. We show that line graphs of trees can be obtained by performing this operation on some graphs, and, as an application, we compute the characteristic polynomials of line graphs of trees.
Consider a connected undirected graph \(G = (V, E)\) and an integer \(r \geq 1\); for any vertex \(v \in V\), let \(B_r(v)\) denote the ball of radius \(r\) centred at \(v\), i.e., the set of all vertices linked to \(v\) by a path of at most \(r\) edges. If for all vertices \(v \in V\), the sets \(B_r(v)\) are different, then we say that \(G\) is \(r\)-twin-free.
In \(r\)-twin-free graphs, we prolong the study of the extremal values that can be reached by some classical parameters in graph theory, and investigate here the maximum degree.
A new construction of authentication codes with arbitration from \((2\nu-2+2+1)\)-dimensional singular pseudo-symplectic geometry on finite fields is given. Assuming that the encoding rules are chosen according to a uniform probability distribution, the parameters and the probabilities of success for different types of deceptions are also computed.
By a defensive alliance in a graph \(G\) we mean any set \(S\) of vertices in \(G\) such that each vertex in \(S\) is adjacent to at least as many vertices inside \(S\), including the vertex itself, as outside \(S\). If, in addition, we require that every vertex outside a defensive alliance \(S\) is adjacent to at least one vertex in \(S\), then \(S\) becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of \(G\). In this paper, we determine bounds for the global defensive alliance numbers in the join, corona, and composition of graphs.
Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). A triangle is a cycle of length three. As usual, \(K_n\) denotes the complete graph on \(n\) vertices. It is shown that for all nonnegative integers \(p\) and \(q\) and for all positive integers \(n\), \(K_n\) can be decomposed into \(p\) copies of \(P_4\) and \(q\) copies of \(C_3\) if and only if \(3(p+q) = e(K_n)\), \(p \neq 1\) if \(n\) is odd, and \(p \geq \frac{n}{2}\) if \(n\) is even.
Motivated by Kotzig and Rosa’s concept of edge magic deficiency, Figueroa-Centeno, Ichishima, and Muntaner-Batle defined a similar concept for super edge magic total labelings. The super edge magic deficiency of a graph \(G\), which is denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) \(+\infty\) if there exists no such \(n\). In this paper, we study the super edge magic deficiency of kite graphs.
The corona of two graphs \(G\) and \(H\), written as \(G \odot H\), is the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and then joining the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae for the Wiener, hyper-Wiener and reverse-Wiener indices of the corona of two graphs.
The energy of a graph \(G\), denoted by \(E(G)\), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of \(G\). Let \(\mathcal{B}(p, q)\) denote the set of bipartite unicyclic graphs with a \((p, q)\)-bipartition, where \(q \geq p \geq 2\). Recently, Li and Zhou [MATCH Commun. Math. Comput. Chem. \(54 (2005) 379-388]\) conjectured that for \(q \geq 3\), \(E(B(3, q)) > E(H(3, q))\), where \(B(3, q)\) and \(H(3, q)\) are respectively graphs as shown in Fig. 1. In this note, we show that this conjecture is true for \(3 \leq q \leq 217\). As a byproduct, we determined the graph with minimal energy among all graphs in \(\mathcal{B}(3, q)\).
In this work, infinite similarities of permutation groups are investigated by means of new methods. For this purpose, we handle distinct groups on the set of natural numbers and we give the separation of the subgroups of them. Afterwards, we give the matrix representation of this groups.
This paper studies edge- and total-colorings of graphs in which (all or only adjacent) vertices are distinguished by their sets of colors. We provide bounds for the minimum number of colors needed for such colorings for the Cartesian product of graphs along with exact results for generalized hypercubes. We also present general bounds for the direct, strong and lexicographic products.
The pebbling number \(f(G)\) of a graph \(G\) is the smallest number \(k\) such that, however \(n\) pebbles are placed on the vertices of \(G\), we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for any connected graphs \(G\) and \(H\), \(f(G \times H) \leq f(G)f(H)\), where \(G \times H\) represents the Cartesian product of \(G\) and \(H\). In this paper, we prove that \(f(G \times H) \leq f(G)f(H)\) when \(G\) has the two-pebbling property and \(H = K_{r,s}^{ – k}\), a graph obtained from the \(r \times s\) complete bipartite graph \(K_{r,s}\) by deleting \(k\) edges which form a matching. We also show that Graham’s conjecture holds for \(K_{r,s}^{-k_1} \times K_{m,n}^{-k_2}\).
The Hosoya polynomial of a graph \(G\) is defined as \(H(G,x) = \sum\limits_{k\geq 0} d(G,k)x^k,\)
where \(d(G, k)\) is the number of vertex pairs at distance \(k\) in \(G\). The calculation of Hosoya polynomials of molecular graphs is a significant topic because some important molecular topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be obtained from Hosoya polynomials. Hosoya polynomials of zig-zag open-ended nanotubes have been given by Xu and Zheng et al. A capped zig-zag nanotube \(T(p, q)[C, D; a]\) consists of a zig-zag open-ended nanotube \(T(p, q)\) and two caps \(C\) and \(D\) with the relative position \(a\) between \(C\) and \(D\). In this paper, we give a general formula for calculating the Hosoya polynomial of any capped zig-zag nanotube. By the formula, the Hosoya polynomial of any capped zig-zag nanotube can be deduced. Furthermore, it is also shown that any two non-isomorphic capped zig-zag nanotubes \(T(p, q)[C, D; a_1]\), \(T(p, q’)[C, D; a_2]\) with \(q’ \geq q^* \geq p+1\) have the same Hosoya polynomial, where \(q^*\) is an integer that depends on the structures of \(C\) and \(D\).
Ewa Wojcicka (Journal of Graph Theory, \(14(1990), 205-215)\) showed that every connected, 3-color-critical graph on more than 6 vertices has a Hamiltonian path. Henning et al. (Discrete Mathematics, \(161(1996), 175-184)\) defined a graph \(G\) to be \(k\)-\((\gamma, d)\)-critical graph if \(\gamma(G) = k\) and \(\gamma(G + uv) = k – 1\) for each pair \(u, v\) of nonadjacent vertices of \(G\) that are at distance at most \(d\) apart. They asked if a 3-\((\gamma, 2)\)-critical graph must contain a dominating path. In this paper, we show that every connected, 3-\((\gamma, 2)\)-critical graph must contain a dominating path. Further, we show that every connected, 3-\((\gamma, 2)\)-critical graph on more than 6 vertices has a Hamiltonian path.
Let \(d(u,v)\) denote the distance between two distinct vertices of a connected graph \(G\) and \(diam(G)\) be the diameter of \(G\). A radio labeling \(f\) of \(G\) is an assignment of positive integers to the vertices of \(G\) satisfying \(d(u,v) + |f(u) – f(v)| \geq diam(G) + 1\). The maximum integer in the range of the labeling is its span. The radio number of \(G\), denoted by \(rn(G)\), is the minimum possible span. In \([7]\) M. Farooq et al. found the lower bound for the radio number of generalized gear graph. In this paper, we give an upper bound for the radio number of generalized gear graph, which coincides with the lower bound found in \([7]\).
In this paper, we study codes over polynomial rings and establish a connection to Jacobi Hilbert modular forms, specifically Hilbert modular forms over the totally real field via the complete weight enumerators of codes over polynomial rings.
For a connected graph \(G\) of order \(3\) or more and an edge coloring \(c: E(G) \to \mathbb{Z}_k\) (\(k \geq 2\)) where adjacent edges may be colored the same, the color sum \(s(v)\) of a vertex \(v\) of \(G\) is the sum in \(\mathbb{Z}_k\) of the colors of the edges incident with \(v\). The edge coloring \(c\) is a modular \(k\)-edge coloring of \(G\) if \(s(u) \neq s(v)\) in \(\mathbb{Z}_k\) for all pairs \(u, v\) of adjacent vertices in \(G\). The modular chromatic index \(\chi_m'(G)\) of \(G\) is the minimum \(k\) for which \(G\) has a modular \(k\)-edge coloring. It is shown that \(\chi(G) \leq \chi_m'(G) \leq \chi(G)+1\) for every connected graph \(G\) of order at least 3, where \(\chi(G)\) is the chromatic number of \(G\). Furthermore, it is shown that \(\chi_m'(G) = \chi(G) + 1\) if and only if \(\chi(G) \equiv 2 \pmod{4}\) and every proper \(\chi(G)\)-coloring of \(G\) results in color classes of odd size.
It is known that an \(\alpha\)-labeling of a bipartite graph \(G\) with \(n\) edges can be used to obtain a cyclic \(G\)-decomposition of \(K_{2nx+1}\) for every positive integer \(x\). It is also known that if two graphs \(G\) and \(H\) admit a free \(a\)-labeling, then their vertex-disjoint union also admits a free \(\alpha\)-labeling. We show that if \(G\) is a bipartite prism, a bipartite Möbius ladder, or a connected cubic bipartite graph of order at most 14, then \(G\) admits a free \(a\)-labeling. We conjecture that every bipartite cubic graph admits a free \(\alpha\)-labeling.
Here we present a characterization of Sheffer-type polynomial sequences based on the isomorphism between the Riordan group and Sheffer group and the sequence characterization of Riordan arrays. We also give several alternative forms of the characterization of the Riordan group, Sheffer group, and their subgroups. Formulas for the computation of the generating functions of Riordan arrays and Sheffer-type polynomial sequences from the characteristics are shown. Furthermore, the applications of the characteristics to lattice walks and recursive construction of Sheffer-type polynomial sequences are also given.
A set of \( S \) edge-disjoint Hamilton cycles in a graph \( G \) is said to be \({maximal}\) if the Hamilton cycles in \( S \) form a subgraph of \( G \) such that \( G – E(S) \) has no Hamilton cycle. The set of integers \( m \) for which a graph \( G \) contains a maximal set of \( m \) edge-disjoint Hamilton cycles has previously been determined whenever \( G \) is a complete graph, a complete bipartite graph, and in many cases when \( G \) is a complete multipartite graph. In this paper, we solve half of the remaining open cases regarding complete multipartite graphs.
For an outerplanar graph on \( n \) vertices, we determine the maximum number of vertices of degree at least \( k \). For \( k = 4 \) (and \( n \geq 7 \)) the answer is \( n – 4 \). For \( k = 5 \) (and \( n \geq 4 \)), the answer is \( \left\lfloor \frac{2(n-4)}{3} \right\rfloor \) (except one less when \( n \equiv 1 \pmod{6} \)). For \( k \geq 6 \) (and \( n \geq k + 2 \)), the answer is \( \left\lfloor \frac{n-6}{k-4} \right\rfloor \). We also determine the maximum sum of the degrees of \( s \) vertices in an \( n \)-vertex outerplanar graph and the maximum sum of the degrees of the vertices with degree at least \( k \).
For a connected graph \( G \) and a positive integer \( k \), the \( k \)th power \( G^k \) of \( G \) is the graph with \( V(G^k) = V(G) \) where \( uv \in E(G^k) \) if the distance \( d_G(u,v) \) between \( u \) and \( v \) is at most \( k \). The edge coloring of \( G^k \) defined by assigning each edge \( uv \) of \( G^k \) the color \( d_G(u,v) \) produces an edge-colored graph \( G^k \) called a distance-colored graph. A distance-colored graph is properly \( p \)-connected if every two distinct vertices \( u \) and \( v \) in the graph are connected by \( p \) internally disjoint properly colored \( u \)-\( v \) paths. It is shown that \( G^2 \) is properly \( 2 \)-connected for every \( 2 \)-connected graph that is not complete, a double star is the only tree \( T \) for which \( T^2 \) is properly \( 2 \)-connected, and \( G^3 \) is properly \( 2 \)-connected for every connected graph \( G \) of diameter at least \( 3 \). All pairs \( k,n \) of positive integers for which \( P_n^k \) is properly \( k \)-connected are determined. It is shown that every properly colored graph \( H \) with \( \chi'(H) \) colors is a subgraph of some distance-colored graph and the question of determining the smallest order of such a graph is studied.
Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). Any edge labeling \( f \) induces a partial vertex labeling \( f^+ : V(G) \to \mathbb{Z}_2 \) assigning \( 0 \) or \( 1 \) to \( f^+(v) \), \( v \) being an element of \( V(G) \), depending on whether there are more \( 0 \)-edges or \( 1 \)-edges incident with \( v \), and no label is given to \( f^+(v) \) otherwise. For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = |\{v \in V(G) : f^+(v) = i\}| \) and let \( e_f(i) = |\{e \in E(G) : f(e) = i\}| \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( |e_f(0) – e_f(1)| \leq 1 \). The edge-balance index set of the graph \( G \) is defined as \( EBI(G) = \{|v_f(0) – v_f(1)| : f \text{ is edge-friendly}\} \). In this paper, exact values of the edge-balance index sets of \( L \)-product of cycles with stars, \( C_n \times_L (S_t(m), c) \), where \( m \) is even, and \( c \) is the center of the star graph are presented.
We investigate group divisible designs with two association classes (known as GDDS, GADs or PBIBDs) with block size 3 and unequal size groups. We completely determine the necessary and sufficient conditions for groups with size vector \((n, 1)\) for any \(n \geq 3\), and \((n, 2, 1)\) for \(n \in \{2, 3, \ldots, 6\}\). We also have some general results for \((n_1, n_2, n_3)\).
A mutation of a vertex-magic total labeling of a graph \( G \) is a swap of some set of edges incident on one vertex of \( G \) with some set of edges incident with another vertex where the labels on the two sets have the same sum. Mutation has previously been seen to be a useful method for producing new labelings from old. In this paper, we study mutations which mutate labelings of regular graphs into labelings of other regular graphs. We present results of extensive computations which confirm how prolific this procedure is. These computations add weight to MacDougall’s conjecture that all non-trivial regular graphs are vertex-magic.
A Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) is a set of \( t \) \( m \)-tuples \( \{(d_{i,1}, d_{i,2}, \ldots, d_{i,m}) \mid i = 1, 2, \ldots, t\} \) such that \( d_{i,1} + d_{i,2} + \cdots + d_{i,m} = 0 \) for \( 1 \leq i \leq t \) and \( \{|d_{i,j}| \mid 1 \leq i \leq t, 1 \leq j \leq m\} = \{d, d+1, \ldots, d+mt-1\} \). In this paper, we give necessary and sufficient conditions on \( t \) and \( d \) for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( m \equiv 0, 2 \pmod{4} \). In the case that \( m \equiv 1, 3 \pmod{4} \), we provide sufficient conditions for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( d \) is at most about \( t/2 \). Using these results, we obtain cyclic \( m \)-cycle systems of the circulant graph \( \langle{d, d+1, \ldots, d+mt-1}\rangle_n \) for all \( n \geq 2(d+mt)-1 \) with \( d \) and \( t \) satisfying certain conditions.
We give a computer-assisted proof of the fact that \( R(K_5 – P_3, K_5) = 25 \). This solves one of the three remaining open cases in Hendry’s table, which listed the Ramsey numbers for pairs of graphs on 5 vertices. We find that there exist no \( (K_5 – P_3, K_5) \)-good graphs containing a \( K_4 \) on 23 or 24 vertices, where a graph \( F \) is \( (G, H) \)-good if \( F \) does not contain \( G \) and the complement of \( F \) does not contain \( H \). The unique \( (K_5 – P_3, K_5) \)-good graph containing a \( K_4 \) on 22 vertices is presented.
A group divisible design (GDD) \( (v = v_1 + v_2 + \cdots + v_g, g, k; \lambda_1, \lambda_2) \) is an ordered pair \( (V, \mathcal{B}) \) where \( V \) is a \( v \)-set of symbols and \( \mathcal{B} \) is a collection of \( k \)-subsets (called blocks) of \( V \) satisfying the following properties: the \( v \)-set is divided into \( g \) groups of sizes \( v_1, v_2, \ldots, v_g \); each pair of symbols from the same group occurs in exactly \( \lambda_1 \) blocks in \( \mathcal{B} \); and each pair of symbols from different groups occurs in exactly \( \lambda_2 \) blocks in \( \mathcal{B} \). In this paper we give necessary conditions on \( m \) and \( n \) for the existence of a \( GDD(v = m+n, 2, 3; 1, 2) \), along with sufficient conditions for each \( m \leq \frac{n}{2} \). Furthermore, we introduce some construction techniques to construct some \( GDD(v = m + n, 2, 3; 1, 2) \)s when \( m > \frac{n}{2} \), namely, a \( GDD(v = 9 + 15, 2, 3; 1, 2) \) and a \( GDD(v = 25 + 33, 2, 3; 1, 2) \).
Let \( D \) be a directed graph. An anti-directed cycle in \( D \) is a set of arcs which form a cycle in the underlying graph, but for which no two consecutive arcs form a directed path in \( D \); this cycle is called an anti-directed Hamilton cycle if it includes all vertices of \( D \). Grant [6] first showed that if \( D \) has even order \( n \), and each vertex indegree and outdegree in \( D \) is a bit more than \( \frac{2n}{3} \), then \( D \) must contain an anti-directed Hamilton cycle. More recently, Busch et al. [1] lowered the lead coefficient, by showing that there must be an anti-directed Hamilton cycle if all indegrees and outdegrees are greater than \( \frac{9n}{16} \), and conjectured that such a cycle must exist if all indegrees and outdegrees are greater than \( \frac{n}{2} \). We prove that conjecture holds for all directed graphs of even order less than 20, and are thus able to extend the above result to show that any digraph \( D \) of even order \( n \) will have an anti-directed Hamilton cycle if all indegrees and outdegrees are greater than \( \frac{11n}{20} \).
Let \( G \) be a \((p,q)\)-graph in which the edges are labeled \( k, k+1, \ldots, k+q-1 \), where \( k \geq 0 \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If the vertex sums are constant, modulo \( p \), then \( G \) is said to be \( k \)-edge-magic. In this paper, we investigate some classes of cubic graphs which are \( k \)-edge-magic. We also provide a counterexample to a conjecture that any cubic graph of order \( p \equiv 2 \pmod{4} \) is \( k \)-edge-magic for all \( k \).
In this paper, we obtain a new set of conditions which are necessary for the existence of balanced arrays of strength eight with two levels by making use of the positive semi-definiteness of the matrix of moments. We also demonstrate, using illustrative examples, that the maximum number of constraints derived using these results are better than those obtained earlier.
A set \( D \subseteq V(G) \) is a dominating set of a graph \( G \) if every vertex of \( G \) not in \( D \) is adjacent to at least one vertex in \( D \). A minimum dominating set of \( G \), also called a \( \gamma(G) \)-set, is a dominating set of \( G \) of minimum cardinality. For each vertex \( v \in V(G) \), we define the domination value of \( v \) to be the number of \( \gamma(G) \)-sets to which \( v \) belongs. In this paper, we find the total number of minimum dominating sets and characterize the domination values for \( P_2 \Box P_n \), and \( P_2 \Box C_n \).
Let \( G \) be the one-point union of two cycles and suppose \( G \) has \( n \) edges. We show via various graph labelings that there exists a cyclic \( G \)-decomposition of \( K_{2nt+1} \) for every positive integer \( t \).
Decompositions of complete or near-complete graphs into spanning trees have been widely studied, but usually in the homogeneous case, where all component trees are isomorphic. A spanning tree decomposition \( \mathcal{T} = (T_1, \ldots, T_n) \) of such a graph is purely heterogeneous if no two trees \( T_i \) are isomorphic. We show existence of such decompositions with the maximum degree condition \( \Delta(T_i) = i+1 \) for each \( i \in [1..n] \), for every largest possible graph of odd order, and every even order graph which is the complement of a spanning tree satisfying a necessary maximum degree condition.
Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces a partial edge labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). For \( i \in \mathbb{Z}_2 \), let \( V_f(i) = \{v \in V(G) : f(v) = i\} \) and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). A labeling \( f \) is called a friendly labeling if \( |V_f(0) – V_f(1)| \leq 1 \). The \( BI(G) \), the balance index set of \( G \), is defined as \( \{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). This paper focuses on the balance index sets of generalized book and ear expansion graphs.
In this paper, we introduce the notion of \((\alpha, \beta)\)-generalized \(d\)-derivations on lattices and investigate some related properties. Also, using the notion of permuting \((\alpha, \beta)\)-triderivation, we characterize the distributive elements of a lattice.
Suppose \(\{P_r\}\) is a nonempty family of paths for \(r \geq 3\), where \(P_r\) is a path on \(r\) vertices. An \(r\)-coloring of a graph \(G\) is said to be \(\{P_r\}\)-free if \(G\) contains no 2-colored subgraph isomorphic to any path \(P_r\) in \(\{P_r\}\). The minimum \(k\) such that \(G\) has a \(\{P_r\}\)-free coloring using \(k\) colors is called the \(\{P_r\}\)-free chromatic number of \(G\) and is denoted by \(\chi_{\{P_r\}}(G)\). If the family \(\{P_r\}\) consists of a single graph \(P_r\), then we use \(\chi_{P_r}(G)\). In this paper, \(\{P_r\}\)-free colorings of Sierpiński-like graphs are considered. In particular, \(\chi_{P_3}(S_n)\), \(\chi_{P_4}(S_n)\), \(\chi_{P_4}(S(n, k))\), \(\chi_{P_3}(S^{++}(n, k))\), and \(\chi_{P_4}(S^{++}(n, k))\) are determined.
Let \(G = (V,E)\) be a graph with \(v = |V(G)|\) vertices and \(e = |E(G)|\) edges. An \((a, d)\)-edge-antimagic total labeling of the graph \(G\) is a one-to-one map \(A\) from \(V(G) \cup E(G)\) onto the integers \(\{1,2,\ldots,v+e\}\) such that the set of edge weights of the graph \(G\), \(W = \{w(xy) : xy \in E(G)\}\) form an arithmetic progression with the initial term \(a\) and common difference \(d\), where \(w(xy) =\lambda(x) + \lambda(y) + \lambda(xy)\) for any \(xy \in E(G)\). If \(\lambda(V(G)) = \{1,2,\ldots,v\}\) then \(G\) is super \((a, d)\)-edge-antimagic total, i.e., \((a,d)\)-EAT. In this paper, for different values of \(d\), we formulate super \((a, d)\)-edge-antimagic total labeling on subdivision of stars \(K_{1,p}\) for \(p \geq 5\).
We discuss the chromaticity of one family of \(K_4\)-homeomorphs which has girth \(7\) and has exactly \(1\) path of length \(1\), and give a sufficient and necessary condition for the graphs in the family to be chromatically unique.
A theta graph is denoted by \(\theta(a,b,c)\), where \(a \leq b \leq c\). It is obtained by subdividing the edges of the multigraph consisting of \(3\) parallel edges \(a\) times, \(b\) times, and \(c\) times each. In this paper, we show that the theta graph is matching unique when \(a \geq 2\) or \(a = 0\), and all theta graphs are matching equivalent when only one of the edges is subdivided one time. We also completely characterize the relation between the largest matching root \(\alpha\) and the length of path \(a, b, c\) of a theta graph, and determine the extremal theta graphs.
The line graph of \(G\), denoted \(L(G)\), is the graph with vertex set \(E(G)\), where vertices \(x\) and \(y\) are adjacent in \(L(G)\) if and only if edges \(x\) and \(y\) share a common vertex in \(G\). In this paper, we determine all graphs \(G\) for which \(L(G)\) is a circulant graph. We will prove that if \(L(G)\) is a circulant, then \(G\) must be one of three graphs: the complete graph \(K_4\), the cycle \(C_n\), or the complete bipartite graph \(K_{a,b}\), for some \(a\) and \(b\) with \(\gcd(a,b) = 1\).
Let \(G\) be a graph. The point arboricity of \(G\), denoted by \(\rho (G)\), is the minimum number of colors that can be used to color the vertices of \(G\) so that each color class induces an acyclic subgraph of \(G\). The list point arboricity \(\rho_l(G)\) is the minimum \(k\) so that there is an acyclic \(L\)-coloring for any list assignment \(L\) of \(G\) which \(|L(v)| \geq k\). So \(\rho(G) \leq \rho_l(G)\). Zhen and Wu conjectured that if \(|V(G)| \leq 3\rho (G)\), then \(\rho_l(G) = p(G)\). Motivated by this, we investigate the list point arboricity of some complete multi-partite graphs of order slightly larger than \(3p(G)\), and obtain \(\rho(K_{m,(1),2(n-1)}) = \rho_l(K_{m(1),2(n-1)})\) \((m = 2,3,4)\).
In this paper, we consider the relationship between toughness and the existence of \([a, b]\)-factors. We obtain that a graph \(G\) has an \([a, b]\)-factor if \(t(G) \geq {a-1} + \frac{a-1}{b}\) with \(b > a > 1\). Furthermore, it is shown that the result is best possible in some sense.
The clique graph of a graph \(G\) is the graph whose vertex set is the set of cliques of \(G\) and two vertices are adjacent if and only if the corresponding cliques have non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. In this paper, we present several results on connected self-clique graphs in which each clique has the same size \(k\) for \(k = 2\) and \(k = 3\).
All parabolic ovals in affine planes of even order \(q \leq 64\) which are preserved by a collineation group isomorphic to \(\mathrm{A\Gamma L}(1,q)\) are determined. They are either parabolas or translation ovals.
We consider the class \({ER}(n, d, \lambda)\) of edge-regular graphs for some \(n > d > \lambda\), i.e., graphs regular of degree \(d\) on \(n\) vertices, with each pair of adjacent vertices having \(\lambda\) common neighbors. It has previously been shown that for such graphs with \(\lambda > 0\) we have \(n \geq 3(d – \lambda)\) and much has been done to characterize such graphs when equality holds.
Here we show that \(n \geq 3(d – \lambda) + 1\) if \(\lambda > 0\) and \(d\) is odd and contribute to the characterization of the graphs in \({ER}(n, d, \lambda)\), \(\lambda > 0\), \(n = 3(d-\lambda)+1\) by proving some lemmas about the structure of such graphs, and by classifying such graphs that satisfy a strong additional requirement, that the number \(t = t(u,v)\) of edges in the subgraph induced by the \(\lambda\) common neighbors of any two adjacent vertices \(u\) and \(v\) is positive, and independent of \(u\) and \(v\). The result is that there are exactly 4 such graphs: \(K_4\) and 3 strongly regular graphs.
If \(G\) is a connected graph, the distance \(d(u, v)\) between two vertices \(u,v \in V(G)\) is the length of a shortest path between them. Let \(W = \{w_1, w_2, \ldots, w_k\}\) be an ordered set of vertices of \(G\) and let \(v\) be a vertex of \(G\). The representation \(r(v|W)\) of \(v\) with respect to \(W\) is the \(k\)-tuple \((d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\). If distinct vertices of \(G\) have distinct representations with respect to \(W\), then \(W\) is called a resolving set or locating set for \(G\). A resolving set of minimum cardinality is called a basis for \(G\) and this cardinality is the metric dimension of \(G\), denoted by \(\dim(G)\).
A family \(\mathcal{G}\) of connected graphs is a family with constant metric dimension if \(\dim(G)\) does not depend upon the choice of \(G\) in \(\mathcal{G}\). In this paper, we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladder \(M_n\) constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder \(M_n\), except when \(n \equiv 2 \pmod{8}\). It is natural to ask for the characterization of regular graphs with constant metric dimension.
In this paper, we obtain an interesting identity by applying two \(g\)-operator identities. From this identity, we can recover the terminating Sears’ \(\prescript{}{3}{\Phi}_2\) transformation formulas and the Dilcher’s identity and the Uchimura’s identity. In addition, an interesting binomial identity can be concluded.
