We give a solution for the intersection problem for disjoint \( 2 \)-flowers in Steiner triple systems.
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 initiate a study of this parameter.
The parity dimension of a graph \( G \) is defined as the dimension of the null space of its closed neighborhood matrix \( N \). A graph with parity dimension \( 0 \) is called all parity realizable (APR). In this paper, a simple recursive procedure for calculating the parity dimension of a tree is given, which is more apt to be used in the context of enumeration than the graph-theoretical characterizations due to Amin, Slater, and Zhang. Applying the recursive relation, we find asymptotic formulas for the number of APR trees and for the average parity dimension of a tree.
The Ramsey multiplicity \( M(G) \) of a graph \( G \) is defined to be the smallest number of monochromatic copies of \( G \) in any two-coloring of edges of \( K_{R(G)} \), where \( R(G) \) is the smallest integer \( n \) such that every graph on \( n \) vertices either contains \( G \) or its complement contains \( G \). With the help of computer algorithms, we obtain the exact values of Ramsey multiplicities for most of isolate-free graphs on five vertices, and establish upper bounds for a few others.
In this paper, we show that the independence polynomial \(I(G^*; x)\) of \(G^*\) is unimodal for any graph \(G^*\) whose skeleton \(G\) has stability number \(\alpha(G) \leq 8\). In addition, we show that the independence polynomial of \(K^*_{2,n}\) is log-concave with a unique mode.
Let \(G = (V,E)\) be a graph. A set \(S \subseteq V\) is a dominating set of \(G\) if every vertex not in \(S\) is adjacent to some vertex in \(S\). The domination number of \(G\), denoted by \(\gamma(G)\), is the minimum cardinality of a dominating set of \(G\). A set \(S \subseteq V\) is a total dominating set of \(G\) if every vertex of \(V\) is adjacent to some vertex in \(S\). The total domination number of \(G\), denoted by \(\gamma_t(G)\), is the minimum cardinality of a total dominating set of \(G\). In this paper, we provide a constructive characterization of those trees with equal domination and total domination numbers.
We consider a variation of a classical Turán-type extremal problem due to Bollobás \([2,p. 398, no. 13]\) as follows: determine the smallest even integer \(\sigma(C^k,n)\) such that every graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(C^k,n)\) has a realization \(G\) containing a cycle with \(k\) chords incident to a vertex on the cycle. Moreover, we also consider a variation of a classical Turán-type extremal result due to Faudree and Schelp \([7]\) as follows: determine the smallest even integer \(\sigma(P_\ell,n)\) such that every graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) \geq \sigma(P_\ell,n)\) has a realization \(G\) containing \(P_\ell\) as a subgraph, where \(P_\ell\) is the path of length 2. In this paper, we determine the values of \(\sigma(P_\ell,n)\) for \(n \geq \ell+1\) and the values of \(\sigma(C^k,n)\) for \(n \geq (k+3)(2k+5)\).
The hyperbolic Fibonacci function, which is the continuous extension of Binet’s formula for the Fibonacci number, transforms the Fibonacci number theory into a “continuous” theory because every identity for the hyperbolic Fibonacci function has its discrete analogy in the framework of the Fibonacci number. In this new paper, we define three important generalizations of the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions and then carry over many concepts and techniques that we learned in a standard setting for the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions to the generalizations of these functions.
A new construction of authentication codes with arbitration from pseudo-symplectic geometry over finite fields is given. The parameters and the probabilities of deceptions of the codes are also computed.
It was conjectured in a recently published paper that for any integer \(k \geq 8\) and any even integer \(n\) with \(2k+3 < n < 2k+\lfloor\frac{k}{2}\rfloor+3\), the \(k\)th power \(C_n^k\) of the \(n\)-cycle is not a divisor graph. In this paper, we prove this conjecture, hence obtaining a complete characterization of those powers of cycles which are divisor graphs.
Inspired by a recent paper by Giulietti, Korchmàros and Torres \([3]\), we provide equations for some quotient curves of the Deligne-Lusztig curve associated to the Suzuki group \(S_z(q)\).
In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation
\[x_{n+1} = \frac{ax_{n-2l}}{b+c\prod\limits_{i=0}^{k+1}x_{n-2i}}, \quad n=0,1,\ldots,\]
where \(a\), \(b\), and \(c\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\) and \(l\) are nonnegative integers, with \(l \leq k+1\).
The chromatic polynomial of a graph \(\Gamma\), \(C(\Gamma; \lambda)\), is the polynomial in \(\lambda\) which counts the number of distinct proper vertex \(\lambda\)-colorings of \(\Gamma\), given \(\lambda\) colors. By applying the addition-contraction method, chromatic polynomials of some sequences of \(2\)-connected graphs satisfy a number of recursive relations. We will show that by knowing the chromatic polynomial of a few small graphs, the chromatic polynomial of each of these sequences can be computed by utilizing either matrices or generating functions.
An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\). A simple graph \(G\) is of \(f\)-class 1 if the \(f\)-chromatic index of \(G\) equals \(\Delta_f(G)\), where \(\Delta_f(G) = \max_{v\in V(G)}\{\left\lceil\frac{d(v)}{f(v)}\right\rceil\}\). In this article, we find a new sufficient condition for a simple graph to be of \(f\)-class 1, which is strictly better than a condition presented by Zhang and Liu in 2008 and is sharp. Combining the previous conclusions with this new condition, we improve a result of Zhang and Liu in 2007.
We provide the specifics of how affine planes of orders three, four, and five can be used to partition the full design comprising all triples on \(9, 16\), and \(25\) elements, respectively. Key results of the approach for order five are generalized to reveal when there is potential for using suitable affine planes of order \(n\) to partition the complete sets of \(n^2\) triples into sets of mutually disjoint triples covering either all \(n^2\), or else precisely \(n^2 – 1\), elements.
In this paper, the notion of left-right and right-left \(f\)-derivation of a BCC-algebra is introduced, and some related properties are investigated. Also, we consider regular \(f\)-derivation and \(d\)-invariant on \(f\)-ideals in BCC-algebras.
Let \(G\) be a planar graph with maximum degree \(\Delta\). It’s proved that if \(\Delta \geq 5\) and \(G\) does not contain \(5\)-cycles and \(6\)-cycles, then \(la(G) = \lceil\frac{\Delta(G)}{2}\rceil\).
We call the digraph \(D\) an \(m\)-coloured digraph if the arcs of \(D\) are coloured with \(m\) colours. A subdigraph \(H\) of \(D\) is called monochromatic if all of its arcs are coloured alike.
A set \(N \subseteq V(D)\) is said to be a kernel by monochromatic paths if it satisfies the following two conditions:
(i) For every pair of different vertices \(u,v \in N\) there is no monochromatic directed path between them.
(ii) For every vertex \(x \in V(D) – N\), there is a vertex \(y \in N\) such that there is an \(xy\)-monochromatic directed path.
In this paper, it is proved that if \(D\) is an \(m\)-coloured \(k\)-partite tournament such that every directed cycle of length \(3\) and every directed cycle of length \(4\) is monochromatic, then \(D\) has a kernel by monochromatic paths.
Some previous results are generalized.
Let \(\mathcal{S}\) be a finite family of sets in \(\mathbb{R}^d\), each a finite union of polyhedral sets at the origin and each having the origin as an extreme point. Fix \(d\) and \(k\), \(0 \leq k \leq d \leq 3\). If every \(d+1\) (not necessarily distinct) members of \(\mathcal{S}\) intersect in a star-shaped set whose kernel is at least \(k\)-dimensional, then \(\cap\{S_i:S_i\in\mathcal{S}\}\) also is a star-shaped set whose kernel is at least \(k\)-dimensional. For \(k\neq 0\), the number \(d+1\) is best possible.
A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The second power of paths \(P_n^2\),is the graph obtained from the path \(P_n\) by adding edges that join all vertices \(u\) and \(v\) with \(d(u,v) = 2\). In this paper, we show that certain combinations of second power of paths, paths, cycles, and stars are cordial. Specifically, we investigate the cordiality of the join and the union of pairs of second power of paths and graphs consisting of one second power of path and one path and one cycle.
We initiate a study of the toughness of infinite graphs by considering a natural generalization of that for finite graphs. After providing general calculation tools, computations are completed for several examples. Avenues for future study are presented, including existence problems for tough-sets and calculations of maximum possible toughness. Several open problems are posed.
Let an \(H\)-point be a vertex of a tiling of \(\mathbb{R}^2\) by regular hexagons of side length 1, and \(D(n)\) a circle of radius \(n\) (\(n \in \mathbb{Z}^+\)) centered at an \(H\)-point. In this paper, we present an algorithm to calculate the number, \(\mathcal{N}_H(D(n))\), of H-points that lie inside or on the boundary of \(D(n)\). Furthermore, we show that the ratio \(\mathcal{N}_H(D(n))/n^2\) tends to \(\frac{2\pi}{S}\) as \(n\) tends to \(\infty\), where \(S = \frac{3\sqrt{3}}{2}\) is the area of the regular hexagonal tiles.
Let \(G\) be a finite, simple graph. We denote by \(\gamma(G)\) the domination number of \(G\). The bondage number of \(G\), denoted by \(b(G)\), is the minimum number of edges of \(G\) whose removal increases the domination number of \(G\). \(C_n\) denotes the cycle of \(n\) vertices. For \(n \geq 5\) and \(n \neq 5k + 3\), the domination number of \(C_5 \times C_n\) was determined in [6]. In this paper, we calculate the domination number of \(C_5 \times C_n\) for \(n = 5k + 3\) (\(k \geq 1\)), and also study the bondage number of this graph, where \(C_5 \times C_n\) is the cartesian product of \(C_5\) and \(C_n\).
A vertex cut that separates the connected graph into components such that every vertex in these components has at least \(g\) neighbors is an \(R^g\)-vertex-cut. \(R^g\)-vertex-connectivity, denoted by \(\kappa^g(G)\), is the cardinality of a minimum \(R^g\)-vertex-cut of \(G\). In this paper, we will determine \(\kappa^g\) and characterize the \(R^g\)-vertex-atom-part for the first and second type Harary graphs.
A graph \(G\) is supereulerian if \(G\) has a spanning eulerian subgraph. We use \(\mathcal{SL}\) to denote the families of supereulerian graphs. In 1995, Zhi-Hong Chen and Hong-Jian Lai presented the following open problem [2, problem 8.8]: Determine
\[L=\min\max\limits_{G\in SL-\{K_1\}}\{\frac{|E(H)|}{|E(G)|} : H \text{ is spanning eulerian subgroup of G}\}.\]
For a graph \(G\), \(O(G)\) denotes the set of all odd-degree vertices of \(G\). Let \(G\) be a simple graph and \(|O(G)| = 2k\). In this note, we show that if \(G\in{SL}\) and \(k \leq 2\), then \(L \geq \frac{2}{3}\).
It is known that the number of Dyck paths is given by a Catalan number. Dyck paths are represented as plane lattice paths which start at the origin \(O\) and end at the point \(P_n = (n,n)\) repeating \((1,0)\) or \((0,1)\) steps without going above the diagonal line \(OP_n\). Therefore, it is reasonable to ask of any positive integers \(a\) and \(b\) what number of lattice paths start at \(O\) and end at point \(A = (a, b)\) repeating the same steps without going above the diagonal line \(OA\). In this article, we show a formula to represent the number of such generalized Dyck paths.
Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f : V(G) \to A\) induces an edge labeling \(f^* : E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \mathrm{card}\{v \in V(G) : f(v) = i\}\) and \(e_f(i) = \mathrm{card}\{e \in E(G) : f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i, j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A\)-friendly if \(|v_f(i)- v_f(j)| \leq 1\) for all \((i, j) \in A \times A\). If \(c(f)\) is a \((0, 1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the friendly index set of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In [13] the friendly index set of cycles are completely determined. In this paper we describe the friendly index sets of cycles with parallel chords. We show that for a cycle with an arbitrary non-empty set of parallel chords, the numbers in its friendly index set form an arithmetic progression with common difference 2.
The eccentricity \(e(v)\) of a vertex \(v\) in a connected graph \(G\) is the distance between \(v\) and a vertex furthest from \(v\). The center \(C(G)\) is the subgraph induced by those vertices whose eccentricity is the radius of \(G\), denoted \(\mathrm{rad}G\), and the periphery \(P(G)\) is the subgraph induced by those vertices with eccentricity equal to the diameter of \(G\), denoted \(\mathrm{diam}G\). The annulus \(\mathrm{Ann}(G)\) is the subgraph induced by those vertices with eccentricities strictly between the radius and diameter of \(G\). In a graph \(G\) where \(\mathrm{rad}G < \mathrm{diam}G\), the interior of \(G\) is the subgraph \(\mathrm{Int}(G)\) induced by the vertices \(v\) with \(e(v) < \mathrm{diam}G\). Otherwise, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Int}(G) = G\). Another subgraph for a connected graph \(G\) with \(\mathrm{rad}G < \mathrm{diam}G\), called the exterior of \(G\), is defined as the subgraph \(\mathrm{Ext}(G)\) induced by the vertices \(v\) with \(\mathrm{rad}G < e(v)\). As with the interior, if \(\mathrm{rad}G = \mathrm{diam}G\), then \(\mathrm{Ext}(G) = G\). In this paper, the annulus, interior, and exterior subgraphs in trees are characterized.
This paper investigates the dihedral group as the array stabilizer of an augmented \(k\)-set of mutually orthogonal Latin squares. Necessary conditions for the stabilizer to be a dihedral group are established. A set of two-variable identities essential for a dihedral group to be contained in an array stabilizer are determined. Infinite classes of models that satisfy the identities are constructed.
A proper vertex coloring of a graph \(G = (V, E)\) is acyclic if \(G\) contains no bicolored cycle. A graph \(G\) is acyclically \(L\)-list colorable if for a given list assignment \(L = \{L(v) : v \in V\}\), there exists a proper acyclic coloring \(\phi\) of \(G\) such that \(\phi(v) \in L(v)\) for all \(v \in V(G)\). If \(G\) is acyclically \(L\)-list colorable for any list assignment with \(|L(v)| = k\) for all \(v \in V\), then \(G\) is acyclically \(k\)-choosable. In this paper, it is proved that every toroidal graph without 4- and 6-cycles is acyclically \(5\)-choosable.
The centro-polyhedral group \(\langle l,m,n\rangle\), for \(l, m, n \in \mathbb{Z}\), is defined by the presentation
\[\langle x, y, z : x^l = y^m = z^n = xyz \rangle.\]
In this paper, we obtain the periods of \(k\)-nacci sequences in centro-polyhedral groups and related groups.
We study two-path convexity in bipartite tournaments. For a bipartite tournament, we obtain both a necessary condition and a sufficient condition on the adjacency matrix for its rank to be two. We then investigate 4-cycles in bipartite tournaments of small rank. We show that every vertex in a bipartite tournament of rank two lies on a four cycle, and bipartite tournaments with a maximum number of 4-cycles do not necessarily have minimum rank.
A set \(S\) of vertices in a graph \(G\) is a clique dominating set of \(G\) if \(S\) contains at least one vertex of every clique \(C\) of \(G\). The clique domination number \(\gamma_q(G)\) and the upper clique domination number \(\gamma_q(G)\) are, respectively, the minimum and maximum cardinalities of a minimal clique dominating set of \(G\). In this paper, we prove that the problem of computing \(\Gamma_q(G)\) is NP-complete even for split graphs and the problem of computing \(\gamma_q(G)\) is NP-complete even for chordal graphs. In addition, for a block graph \(BG\) we show that the clique domination number is bounded above by the vertex independence number (\(\gamma_q(BG) \leq \beta(BG)\)) and give a linear algorithm for computing \(\gamma_q(BG)\).
Katerinis established the following result in [1]. Let \(G\) be a simple graph with \(\delta(G) \geq \lfloor\frac{|V(G)|}{2}+k\), where \(k\) is a non-negative integer. Let \(f : V(G) \to \mathbb{Z}^+\) be a function having the following properties:
(1) \(\frac{1}{2}\left({d_G(v) – (k+1)}{2}\right) \leq f(v) \leq \frac{1}{2}\left({d_G(v) + (k+1)}{2}\right)\) for every \(v \in V(G)\),
(2) \(\sum_{v\in V(G)} f(v) = |E(G)|\).
Then \(G\) has an orientation \(D\) such that \(d^+_D(v) = f(v)\), for every \(v \in V(G)\). In this paper, we focus on the sharpness of the above two inequalities.
A cosimple regular matroid \(M\) does not have disjoint circuits if and only if \(M \in \{M(K_{3,3}), M^*(K_n) \mid n \geq 3\}\). This extends a former result of Erdős and Pósa on graphs without disjoint circuits.
Suppose \(G\) is a finite graph with vertex-set \(V(G)\) and edge-set \(E(G)\). An \((a, d)\)-edge-antimagic total labeling on \(G\) is a one-to-one map \(f\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G)| + |E(G)|\) with the property that the edge-weights \(w(uv) = f(u) + f(v) + f(uv)\), \(uv \in E(G)\), form an arithmetic progression starting from \(a\) and having common difference \(d\). Such a labeling is called super if the smallest labels appear on the vertices. In this paper, we investigate the existence of super \((a, d)\)-edge-antimagic total labelings of disjoint union of multiple copies of complete bipartite graph.
Let \(G = (V, E)\) be a graph with no isolated vertex. A classical observation in domination theory is that if \(D\) is a minimum dominating set of \(G\), then \(V \setminus D\) is also a dominating set of \(G\). A set \(D’\) is an inverse dominating set of \(G\) if \(D’\) is a dominating set of \(G\) and \(D’ \subseteq V \setminus D\) for some minimum dominating set \(D\) of \(G\). The inverse domination number of \(G\) is the minimum cardinality among all inverse dominating sets of \(G\). The independence number of \(G\) is the maximum cardinality of an independent set of vertices in \(G\). Domke, Dunbar, and Markus (Ars Combin. \(72 (2004), 149-160)\) conjectured that the inverse domination number of \(G\) is at most the independence number of \(G\). We prove this conjecture for special families of graphs, including claw-free graphs, bipartite graphs, split graphs, very well covered graphs, chordal graphs, and cactus graphs.
A \(\lambda\)-design on \(v\) points is a set of \(v\) distinct subsets (blocks) of a \(v\)-set such that any two different blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that every \(\lambda\)-design can be obtained from a symmetric design by a certain complementation procedure. A result of Ryser and Woodall establishes that there exist two integers, \(r\) and \(r^*\), such that each point in a \(\lambda\)-design is in exactly \(r\) or \(r^*\) blocks. The main result of the present paper is that the \(\lambda\)-design conjecture is true for \(\lambda\)-designs with \(\gcd(r-1,r^*-1)=7\).
Improving on Domokos’s improvement of Swan’s theorem, we show that under certain conditions on a finite digraph, whenever \(p,q\) are vertices, then the number of even Eulerian paths from \(p\) to \(q\) is the same as the number of odd ones from \(p\) to \(q\).
Double-loop networks have been widely studied as architecture for local area networks. A double-loop network \(G(N;s_1,s_2)\) is a digraph with \(N\) vertices \(0,1,\ldots,N-1\) and \(2N\) edges of two types:
\(s_1-edge\): \(i \rightarrow i+s_1 \pmod{N}\); \(i=0,1,\ldots,N-1\).
\(s_2-edge\): \(i \rightarrow i+s_2 \pmod{N}\); \(i=0,1,\ldots,N-1\).
for some fixed steps \(1 \leq s_1 < s_2 < N\) with \(\gcd(N,s_1,s_2) = 1\). Let \(D(N;s_1,s_2)\) be the diameter of \(G\) and let us define \(D(N) = \min\{D(N;s_1,s_2) | 1 \leq s_1 < s_2 < N \text{ and } gcd(N,s_1,s_2) = 1\}\), and \(D_1(N) = \min\{D(N;1,s) | 1 < s < N\}\). If \(N\) is a positive integer and \(D(N) < D_1(N)\), then \(N\) is called a non-unit step integer or a nus integer. Xu and Aguild et al. gave some infinite families of 0-tight nus integers with \(D_1(N) – D(N) \geq 1\). In this work, we give a method for finding infinite families of nus integers. As application examples, we give one infinite family of 0-tight nus integers with \(D_1(N) – D(N) \geq 5\), one infinite family of 2-tight nus integers with \(D_1(N) – D(N) \geq 1\) and one infinite family of 3-tight nus integers with \(D_1(N) – D(N) \geq 1\).
Ramanujan’s \(\mathop{_1\psi_1}\)-summation formula is one of the fundamental identities in basic hypergeometric series. We review proofs of this identity and clarify its connections with other basic hypergeometric series transformations and formulae. In particular, we shall put our main emphasis on methods that can be used not only to provide deeper insight into Ramanujan’s \(\mathop{_1\psi_1}\)-summation formula, but also to derive new transformations and identities for basic hypergeometric series.
A diagonalised lattice is a two dimensional grid, where we add exactly one arbitrary diagonal in each square, and color each vertex black or white.We show that for every diagonalised lattice there is a walk from the left to the right, using only black vertices, if and only if there is no walk from the top to the bottom, using only white vertices.
A terrace for \(\mathbb{Z}_m\) is an arrangement \((a_1, a_2, \ldots, a_m)\) of the \(m\) elements of \(\mathbb{Z}_m\) such that the sets of differences \(a_{i+1} – a_i\) and \(a_i – a_{i+1}\) (\(i = 1, 2, \ldots, m-1\)) between them contain each element of \(\mathbb{Z}_m \setminus \{0\}\) exactly twice. For \(m\) odd, many procedures are available for constructing power-sequence terraces for \(\mathbb{Z}_m\); each such terrace may be partitioned into segments one of which contains merely the zero element of \(\mathbb{Z}_m\) whereas each other segment is either (a) a sequence of successive powers of a non-zero element of \(\mathbb{Z}_m\) or (b) such a sequence multiplied throughout by a constant. For \(n\) an odd prime power satisfying \(n \equiv 1\) or \(3 \pmod{8}\), this idea has previously been extended by using power-sequences in \(\mathbb{Z}_n\) to produce some \(\mathbb{Z}_m\) terraces \((a_1, a_2, \ldots, a_m)\) where \(m = n+1 = 2^\mu\), with \(a_{i+1} – a_i = -(a_{i+1+\mu} – a_{i+\mu})\) for all \(i \in [1, \mu-1]\). Each of these “da capo directed terraces” consists of a sequence of segments, one containing just the element \(0\) and another just containing the element \(n\), the remaining segments each being of type (a) or (b) above with each of its distinct entries \(z\) from \(\mathbb{Z}_n \setminus \{0\}\) evaluated so that \(1 \leq x \leq n-1\). Now, for many odd prime powers \(n\) satisfying \(n \equiv 1 \pmod{4}\), we similarly produce narcissistic terraces for \(\mathbb{Z}_{n+1}\); these have \(a_{i+1} – a_i = a_{m-i+1} – a_{m-i}\) for all \(i \in [1, \mu-1]\).
We determine the full Sylow \(p\)-subgroup of the automorphism group of transitive \(k\)-ary relational structures of order \(p^2\), \(p\) a prime. We then find the full automorphism group of transitive ternary relational structures of order \(p^2\), for those values of \(p\) for which \({A_p}\) is the only doubly-transitive nonabelian simple group of degree \(p\). Finally, we determine optimal necessary and sufficient conditions for two Cayley \(k\)-ary relational structures of order \(p^2\), \(k < p\), to be isomorphic.
Let \(G\) be a finite group, \(S\) (possibly, contains the identity element) be a subset of \(G\). The Bi-Cayley graph \(\text{BC}(G, S)\) is a bipartite graph with vertex set \(G \times \{0,1\}\) and edge set \(\{\{(g,0), (gs,1)\}, g \in G, s \in S\}\). A graph \(X\) is said to be super-edge-connected if every minimum edge cut of \(X\) is a set of edges incident with some vertex. The restricted edge connectivity \(\lambda'(X)\) of \(X\) is the minimum number of edges whose removal disconnects \(X\) into nontrivial components. A \(k\)-regular graph \(X\) is said to be optimally super-edge-connected if \(X\) is super-edge-connected and its restricted edge connectivity attains the maximum \(2k-2\). In this paper, we show that all connected Bi-Cayley graphs, except even cycles, are optimally super-edge-connected.
A transitive triple, \((a, b, c)\), is defined to be the set \(\{(a, b), (b, c), (a, c)\}\) of ordered pairs. A directed triple system of order \(v\), \(DTS(v)\), is a pair \((D, \beta)\), where \(D\) is a set of \(v\) points and \(\beta\) is a collection of transitive triples of pairwise distinct points of \(D\) such that any ordered pair of distinct points of \(D\) is contained in precisely one transitive triple of \(\beta\). An antiautomorphism of a directed triple system, \((D, \beta)\), is a permutation of \(D\) which maps \(\beta\) to \(\beta^{-1}\), where \(\beta^{-1} = \{(c, b, a) | (a, b, c) \in \beta\}\). In this paper, we give necessary and sufficient conditions for the existence of a directed triple system of order \(v\) admitting an antiautomorphism consisting of two cycles of lengths \(M\) and \(2M\), and one fixed point.
A \(k\)-circuit is a directed cycle of length \(k\). In this paper, we completely solve the problem of finding maximum packings and minimum coverings of \(\lambda\)-fold complete bipartite symmetric digraphs with \(6\)-circuits.
Until now, all known simple \(t-(v, k, \lambda)\) designs with \(t \geq 6\) have \(\lambda \geq 4\). On the other hand, P. J. Cameron and C. E. Praeger showed that there are no flag-transitive simple \(7-(v, k, \lambda)\) designs. In the present paper, we considered the flag-transitive simple \(6-(v, k, \lambda)\) designs and proved that there are no non-trivial flag-transitive simple \(6-(v, k, \lambda)\) designs with \(\lambda \leq 5\).
In \(1972\), Erdős, Faber, and Lovász made the now famous conjecture: If a graph \(G\) consists of \(n\) copies of the complete graph \(K_n\), such that any two copies have at most one common vertex (such graphs are called EFL graphs), then \(G\) is \(n\)-colorable. In this paper, we show that the conjecture is true for two different classes of EFL graphs. Furthermore, a new shorter proof of the conjecture is given for a third class of EFL graphs.
The edge set of \(K_n\) cannot be decomposed into edge-disjoint octagons (or \(8\)-cycles) when \(n \not\equiv 1 \pmod{16}\). We consider the problem of removing edges from the edge set of \(K_n\) so that the remaining graph can be decomposed into edge-disjoint octagons. This paper gives the solution of finding maximum packings of complete graphs with edge-disjoint octagons and the minimum leaves are given.
In this paper, we introduced the notion of symmetric \(f\) bi-derivations on lattices and investigated some related properties. We characterized the distributive lattice by symmetric \(f\) bi-derivations.
In \(1990\), Anderson et al. \([1]\) generalized the competition graph of a digraph to the competition multigraph of a digraph and defined the multicompetition number of a multigraph. The competition multigraph \(CM(D)\) of a digraph \(D = (V, A)\) is the multigraph \(M = (V, E’)\) where two vertices of \(V\) are joined by \(k\) parallel edges if and only if they have exactly \(\ell\) common preys in \(D\). The multicompetition number \(k^*(M)\) of the multigraph \(M\) is the minimum number \(p\) such that \(M \cup I_p\) is the competition multigraph of an acyclic digraph, where \(I_k\) is a set of \(p\) isolated vertices. In this paper, we study the multicompetition numbers for some multigraphs and generalize some results provided by Kim and Roberts \([9]\), and by Zhao and He \([18]\) on general competition graphs, respectively.
Frank Harary contributed numerous questions to a variety of topics in graph theory. One of his favourite topics was the Reconstruction Problem, which, in its first issue in \(1977\), the Journal of Graph Theory described as the major unsolved problem in the field. Together with Plantholt, Frank Harary initiated the study of reconstruction numbers of graphs. We shall here present a survey of some of the work done on reconstruction numbers, focusing mainly on the questions which this work leaves open.
An upper bound of the basis number of the lexicographic product of two graphs from the basis number of the factors is presented. Furthermore, the basis numbers of the lexicographic product of some classes of graphs is determined.
In this paper, we prove that for any graph \(G\), \(\lambda(G^{+++}) = \delta(G^{-++})\) and all but for a few exceptions, \(G^{-++}\) is super edge-connectivity where \(G^{-++}\) is the transformation graph of a graph \(G\) introduced in \([1]\).
A graph-pair of order \(t\) is two non-isomorphic graphs \(G\) and \(H\) on \(t\) non-isolated vertices for which \(G \cup H \cong K_t\) for some integer \(t \geq 4\). Given a graph-pair \((G,H)\), we say \((G, H)\) divides some graph \(K\) if the edges of \(K\) can be partitioned into copies of \(G\) and \(H\) with at least one copy of \(G\) and at least one copy of \(H\). We will refer to this partition as a \((G, H)\)-\(multidecomposition\) of \(K\).
In this paper, we consider the existence of multidecompositions of \(K_n – F\) into graph-pairs of order \(5\) where \(F\) is a Hamiltonian cycle or (almost) \(1\)-factor.
The upper chromatic number \(\overline{\chi}_u(\mathcal{H})\) of a \(C\)-hypergraph \(\mathcal{H} = (X, C)\) is the maximum number of colors that can be assigned to the vertices of \(\mathcal{H}\) in such a way that each \(C \in \mathcal{C}\) contains at least a monochromatic pair of vertices. This paper gives an upper bound for the upper chromatic number of Steiner triple systems of order \(n\) and proves that it is best possible for any \(n (\equiv 1 \text{ or } 3 \pmod{6})\).
A map is called Unicursal if it has exactly two vertices of odd valency. A near-triangulation is a map with all but one of its faces triangles. We use the enufunction approach to enumerate rooted Unicursal planar near-triangulations with the valency of the root-face and the number of non-rooted faces as parameters.
An edge coloring is proper if no two adjacent edges are assigned the same color and vertex-distinguishing proper coloring if it is proper and incident edge sets of every two distinct vertices are assigned different sets of colors. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph \(G\) is denoted by \(\overline{\chi}'(G)\). In this paper, we prove that \(\overline{\chi}'(G) \leq \Delta(G) + {4}\) if \(G = (V, E)\) is a connected graph of order \(n \geq 3\) and \(\sigma_2(G) \geq n\), where \(\sigma_2(G) = \min\{d(x) + d(y) | xy \in E(G)\}\).
In this paper, we study the minimum distance between the set of bent functions and the set of \(1\)-resilient Boolean functions and present lower bounds on that. The first bound is proved to be tight for functions up to \(10\) input variables and a revised bound is proved to be tight for functions up to \(14\) variables. As a consequence, we present a strategy to modify the bent functions, by toggling some of its outputs, in getting a large class of \(1\)-resilient functions with very good nonlinearity and autocorrelation. In particular, the technique is applied up to \(14\)-variable functions and we show that the construction provides a large class of \(1\)-resilient functions reaching currently best known nonlinearity and achieving very low autocorrelation values which were not known earlier. The technique is sound enough to theoretically solve some of the mysteries of \(8\)-variable, \(1\)-resilient functions with maximum possible nonlinearity. However, the situation becomes complicated from \(10\) variables and above, where we need to go for complicated combinatorial analysis with trial and error using computational facility.
In this paper, we characterize the variety of quasi-groups isotopic to abelian groups by four-variable identities.
A direct method for constructing large sets of \(t\)-designs is based on the concept of assembling orbits of a permutation group \(G\) on \(k\)-subsets of a \(v\)-set into block sets of \(t\)-designs so that these designs form a large set. If \(G\) is \(t\)-homogeneous, then any orbit is a \(t\)-design and therefore we obtain a large set by partitioning the set of orbits into parts consisting of the same number of \(k\)-subsets. In general, it is hard to find such partitions. We solve this problem when orbit sizes are limited to two values. We then use its corollaries to obtain some results in a special case in which a simple divisibility condition holds and no knowledge about orbit sizes is assumed.
Dean \(([3])\) shows that if \(G\) be a \(k\)-connected graph such that any fragment whose neighborhood contains an edge has cardinality exceeding \(\frac{k}{2}\), then the subgraph \(H = (V(G), E_k(G))\) formed by \(V(G)\) and the \(k\)-contractible edges of \(G\) is \(2\)-connected. In this paper, we show that for \(k = 4\), Dean’s result holds when reduced \(\frac{k}{2}\) to \(\frac{k}{4}\). But for \(k \geq 5\), we give a counterexample to show that it is false and give a lower bound of the number of \(k\)-contractible edges for \(k = 5\).
Let \(G\) be a finite simple \(\chi\)-chromatic graph and \(\mathcal{L} = \{L_u\}_{u\in V(G)}\) be a list assignment to its vertices with \(L_u \subseteq \{1,…,\chi\}\). A list colouring problem \((G, \mathcal{L})\) with a unique solution for which the sum \(\sum_{u\in V(G)}|L_u|\) is maximized, is called a \(maximum\; \chi-list \;assignment\) of \(G\). In this paper, we prove a \(Circuit\; Simulation\) Lemma that, strictly speaking, makes it possible to simulate any Boolean function by \(effective\) 3-colourings of a graph that is \(polynomial-time \;constructable\) from the Boolean function itself. We use the lemma to simply prove some old results as corollaries, and also we prove that the following decision problem, related to the computation of the fixing number of a graph [Daneshgar \(1997\), Daneshgar and Naserasr, Ars Combin. \(69\) \((2003)\)], is \(\sum_{2}^{P}\)-complete.
In this paper, we give another proof for labeled spanning forests of the complete bipartite graph \(K_{m,n}\), and obtain two Abel-type polynomials. And then we investigate the enumeration of non-trivial rooted labeled spanning forests of the complete bipartite graph \(K_{m,n}\).
In this paper, we investigate the global behavior of the difference equation
\[x_{n+1} = \frac{\alpha x_{n-k}}{\beta+\gamma x_{n-(k+1)}^p},\text{n=1,2,}\ldots\]
with non-negative parameters and non-negative initial conditions, where \(k\) is an odd number.
In many papers, the relation between the domination number of a product of graphs and the product of domination numbers of factors is studied. Here we investigate this problem for domination and total domination numbers in the cross product of digraphs. We give analogues of known results for graphs, and we also present new results for digraphs with sources. Using these results, we find domination (total domination) numbers for some classes of digraphs.
Let \(P_{k+1}\) denote a path of length \(k\) and let \(C_k\) denote a cycle of length \(k\). As usual, \(K_n\) denotes the complete graph on \(n\) vertices. In this paper, we investigate decompositions of \(K_n\) into paths and cycles, and give some necessary and/or sufficient conditions for such a decomposition to exist. Besides, we obtain a necessary and sufficient condition for decomposing \(K_n\) into \(p\) copies of \(P_5\) and \(q\) copies of \(C_4\) for all possible values of \(p\geq 0\) and \(q\geq 0\).
Given a simple connected undirected graph \(G\), the Wiener index \(W(G)\) of \(G\) is defined as half the sum of the distances over all pairs of vertices of \(G\). In practice, \(G\) corresponds to what is known as the molecular graph of an organic compound. We obtain a sharp lower bound for \(W(G)\) of an arbitrary graph in terms of the order, size, and diameter of \(G\).
The Zagreb indices are topological indices of graphs, which are defined as:\(M_1(G) = \sum\limits_{v \in V(G)} (d(v))^2\), \(M_2(G) = \sum\limits_{uv \in E(G)} d(u)d(v)\) .In this paper, we determine the upper and lower bounds for the Zagreb indices of unicyclic graphs in terms of their order and girth. In each case, we characterize the extremal graphs.
In this paper, we are concerned with Leibniz numbers. We establish a series of identities involving Leibniz numbers, Stirling numbers, harmonic numbers, arctan numbers by making use of generating functions. In addition, we give the asymptotic expansion of certain sums related to Leibniz numbers by Laplace’s method.
We consider the undirected simple connected graph for which edges fail independently of each other with equal probability \(1 – p\) and nodes are perfect. The all-terminal reliability of a graph \(G\) is the probability that the spanning subgraph of surviving edges is connected, denoted as \(R(G,p)\). Graph \(G \in \Omega(n,e)\) is said to be uniformly least reliable if \(R(G,p) \leq R(G’,p)\) for all \(G’ \in \Omega(n,e)\), and for all edge failure probabilities \(0 < 1 – p < 1\). In this paper, we prove the existence of uniformly least reliable graphs in the class \(\Omega(n,e)\) for \(e \leq n + 1\) and give their topologies.
We study V- and \(\Lambda\)-patterns which generalize valleys and peaks, as well as increasing and decreasing runs, in permutations. A complete classification of permutations (multi)-avoiding V- and \(\Lambda\)-patterns of length \(4\) is given. We also establish a connection between restricted permutations and matchings in the coronas of complete graphs.
Let \(G\) be a connected graph. A weakly connected dominating set of \(G\) is a dominating set \(D\) such that the edges not incident to any vertex in \(D\) do not separate the graph \(G\). In this paper, we first consider the relationship between weakly connected domination number \(\gamma_w(G)\) and the irredundance number \(ir(G)\). We prove that \(\gamma_w(G) \leq \frac{5}{2}ir(G) – 2\) and this bound is sharp. Furthermore, for a tree \(T\), we give a sufficient and necessary condition for \(\gamma_c(T) = \gamma_w(T) + k\), where \(\gamma_c(T)\) is the connected domination number and \(0 \leq k \leq \gamma_w(T) – 1\).
For two vertices \(u\) and \(v\) in a strong digraph \(D\), the strong distance \(sd(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). The strong eccentricity \(se(v)\) of a vertex \(v\) of \(D\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(srad(D)\) (resp. strong diameter \(sdiam(D)\)) of \(D\) is the minimum (resp. maximum) strong eccentricity among all vertices of \(D\). The lower (resp. upper) orientable strong radius \(srad(G)\) (resp. \(SRAD(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong radius over all strong orientations of \(G\). The lower (resp. upper) orientable strong diameter \(sdiam(G)\) (resp. \(SDIAM(G)\)) of a graph \(G\) is the minimum (resp. maximum) strong diameter over all strong orientations of \(G\). In this paper, we determine the lower orientable strong radius and strong diameter of the Cartesian product of complete graphs, and give the upper orientable strong diameter and the bounds on the upper orientable strong radius of the Cartesian product of complete graphs.
In this paper, we show that the disjoint union of two cordial graphs, one of them is of even size, is cordial and the join of two cordial graphs, both are of even size or one of them is of even size and one of them is of even order, is cordial. We also show that \(C_m \cup C_n \) is cordial if and only if \(m+n \not\equiv 2 \pmod{4}\) and \(mC_n\) is cordial if and only if \(mn \not\equiv 2 \pmod{4}\) and for \(m, n \geq 3\), \(C_m + C_n\) is cordial if and only if \((m, n) \neq (3, 3)\) and \(\{m, n\} \not\equiv \{0, 2\} \pmod{4}\).
Finally, we discuss the cordiality of \(P_n^k\).
We show that a finite linear space with \(b = n^2 + n + 1\) lines, \(n \geq 2\), constant point-degree \(n+1\) and containing a sufficient number of lines of size \(n\) can be embedded in a projective plane of order \(n\). Using this fact, we also give characterizations of some pseudo-complements, which are the complements of certain subsets of finite projective planes.
The numbers of distinct self-orthogonal Latin squares (SOLS) and idempotent SOLS have been enumerated for orders up to and including $9$. The isomorphism classes of idempotent SOLS have also been enumerated for these orders. However, the enumeration of the isomorphism classes of non-idempotent SOLS is still an open problem. By utilising the automorphism groups of class representatives from the already enumerated isomorphism classes of idempotent SOLS, we enumerate the isomorphism classes of non-idempotent SOLS implicitly (i.e. without generating them). New symmetry classes of SOLS are also introduced, based on the number of allowable transformations that may be applied to a SOLS without destroying the property of self-orthogonality, and these classes are also enumerated.
The supereulerian index of a graph \(G\) is the smallest integer \(k\) such that the \(k\)-th iterated line graph of \(G\) is supereulerian. We first show that adding an edge between two vertices with degree sums at least three in a graph cannot increase its supereulerian index. We use this result to prove that the supereulerian index of a graph \(G\) will not be changed after either of contracting an \(A_G(F)\)-contractible subgraph \(F\) of a graph \(G\) and performing the closure operation on \(G\) (if \(G\) is claw-free). Our results extend Catlin’s remarkable theorem \([4]\) relating that the supereulericity of a graph is stable under the contraction of a collapsible subgraph.
This paper is based on the splitting operation for binary matroids that was introduced by Raghunathan, Shikare and Waphare [ Discrete Math. \(184 (1998)\), p.\(267-271\)] as a natural generalization of the corresponding operation in graphs. Here, we consider the problem of determining precisely which graphs \(G\) have the property that the splitting operation, by every pair of edges, on the cycle matroid \(M(G)\) yields a graphic matroid. This problem is solved by proving that there are exactly four minor-minimal graphs that do not have this property.
In this paper, we study the connection of number theory with graph theory via investigating some uncharted properties of the directed graph \(\Gamma'(n)\) whose vertex set is \(\mathbb{Z}_n = \{0,1,\ldots,n-1\}\), and for which there is a directed edge from \(a \in \mathbb{Z}_n\) to \(b \in \mathbb{Z}_n\) if and only if \(a^3 \equiv b \pmod{n}\). For an arbitrary prime \(p\), the formula for the decomposition of the graph \(\Gamma(p)\) is established. We specify two subgraphs \(\Gamma_1(n)\) and \(\Gamma_2(n)\) of \(\Gamma(n)\). Let \(\Gamma_1(n)\) be induced by the vertices which are coprime to \(n\) and \(\Gamma_2(n)\) by induced by the set of vertices which are not coprime to \(n\). We determine the level of every component of \(\Gamma_1(n)\), and establish necessary and sufficient conditions when \(\Gamma_1(n)\) or \(\Gamma_2(n)\) has no cycles with length greater than \(1\), respectively. Moreover, the conditions for the semiregularity of \(\Gamma_2(n)\) are presented.
We made a computer search for minimal blocking sets in the projective geometry \(\text{PG}(2,11)\), and found \(30,000\), of which only two nontrivial blocking sets had the possibility of being isomorphic.
A graph \(G\) is called \({claw-free}\) if \(G\) has no induced subgraph isomorphic to \(K_{1,3}\). Ando et al. obtained the result: a claw-free graph \(G\) with minimum degree at least \(d\) has a path-factor such that the order of each path is at least \(d+1\); in particular \(G\) has a \(\{P_3, P_4, P_5\}\)-factor whenever \(d \geq 2\). Kawarabayashi et al. proved that every \(2\)-connected cubic graph has a \(\{P_3, P_4\}\)-factor. In this article, we show that if \(G\) is a connected claw-free graph with at least \(6\) vertices and minimum degree at least \(2\), then \(G\) has a \(\{P_3, P_4\}\)-factor. As an immediate consequence, it follows that every claw-free cubic graph (not necessarily connected) has a \(\{P_3, P_4\}\)-factor.
In this paper, we study the combinatorial properties of \(w\)-IPP (identifiable parents property) codes and give necessary and sufficient conditions for a code to be a \(w\)-IPP code. Furthermore, let \(R(C) = \frac{1}{n}{\log_q|C|}\) denote the rate of the \(q\)-ary code \(C\) of length \(n\), suppose \(q \geq 3\) is a prime power, we prove that there exists a sequence of linear \(q\)-ary \(2\)-IPP codes \(C_n\) of length \(n\) with \(R(C_n) = \frac{1}{3}log\frac{q^3}{4q^2-6q+3}\).
Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent, denoted \(G \sim H\), if \(P(G,\lambda) = P(H,\lambda)\). We write \([G] = \{H | H \sim G\}\). If \([G] = \{G\}\), then \(G\) is said to be chromatically unique. In this paper, we first characterize certain complete tripartite graphs \(G\) according to the number of \(4\)-independent partitions of \(G\). Using these results, we investigate the chromaticity of \(G\) with certain star or matching deleted. As a by-product, we obtain new families of chromatically unique complete tripartite graphs with certain star or matching deleted.
In this paper, we introduce a hyperoperation associated to the set of all arithmetic functions and analyze the properties of this new hyperoperation. Several characterization theorems are obtained, especially in connection with multiplicative functions.
In this paper, we introduce two new types of labelings of graphs using Fibonacci numbers, namely, Fibonacci graceful labelings and Super Fibonacci graceful labelings. We discuss the existence and non-existence of Fibonacci and Super Fibonacci graceful labelings for certain classes of graphs. Also, we discuss the Fibonacci gracefulness of disjoint union of Super Fibonacci graceful graphs, pendant edge extension of Super Fibonacci graceful graphs, and amalgamation of Super Fibonacci graceful graphs. Finally, we compare the graceful graphs with Fibonacci graceful graphs.
Let the columns of a \(p \times q\) matrix \(M\) over any ring be partitioned into \(n\) blocks, \(M = [M_1, \ldots, M_n]\). If no \(p \times p\) submatrix of \(M\) with columns from distinct blocks \(M_{i}\) is invertible, then there is an invertible \(p \times p\) matrix \(Q\) and a positive integer \(m \leq p\) such that \([QM_1, \ldots, QM_n]\) is in reduced echelon form and in all but at most \(m – 1\) blocks \(QM_i\) the last \(m\) entries of each column are either all zero or they include a non-zero non-unit.
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 largest numbers of maximal independent sets among all quasi-tree graphs and quasi-forest graphs. We also characterize those extremal graphs achieving these values.
In \(2004\), Fischermann et al. \([2]\) generalized bound polysemy to competition polysemy by using digraphs instead of posets. They provided a characterization of competition polysemic pairs and a characterization of the connected graphs \(G\) for which there exists a tree \(T\) such that \((G,T)\) is competition polysemic. In this paper, we continue to study the competition polysemy and characterize the connected graphs \(G\) for which there exists a triangle-free unicyclic graph \(G’\) such that \((G,G’)\) is competition polysemic. Furthermore,we generalize competition polysemy to \(m\)-competition polysemy and
prove a characterization of \(m\)-competition polysemic pairs.
A diagonally switchable \(\lambda\)-fold \(4\)-cycle system of order \(n\), briefly DS4CS\((n, \lambda)\), is a \(\lambda\)-fold \(4\)-cycle system in which by replacing each \(4\)-cycle \((a,b,c,d)\) covering pairs \(ab, bc, cd, da\) by either of the \(4\)-cycles \((a,c,b,d)\) or \((a,d,c,b)\) another \(\lambda\)-fold \(4\)-cycle system is obtained. In \([3]\) Adams, Bryant, Grannell, and Griggs proved that a DS4CS\((n, 1)\) exists if and only if \(n \equiv 1 \pmod{8}\), \(n \geq 17\) with the possible exception of \(n = 17\). In this paper we prove that for \(\lambda \geq 2\) the necessary conditions for the existence of a \(A\)-fold \(4\)-cycle system of order \(7\) are also sufficient for the existence of a DS4CS\((n, \lambda)\) except for \((n, \lambda) = (5, 2)\).
In this paper, we consider the relationships between the sums of the generalized order-\(k\) Fibonacci and Lucas numbers and \(1\)-factors of bipartite graphs.
The ATSP polytope can be expressed by an asymmetric polynomial-size linear program.
A model that represents the rate of changes of the population with limited environmental resources can be described by,
\[
\frac{dp}{dt} = p\left(a – {bp}\right) + g(t,p) = p(t_0)= p_0
\]
where \( a \) measures the growth rate in the absence of the restriction force \( b \) and \( \frac{a}{b} \) is called the carrying capacity of the environment. The random perturbation \( g(t,P) \) is generated by random change in the environment. The behavior of the solution of this model for continuous and discrete case when \( g(t,P)=w(t) \) is density independent with a constant random factor \( w \) in a short time interval \([t, t + \delta t)\) will be studied. The stability and the behavior of the equilibrium point will also be investigated. A computational approach to the solution using Excel spreadsheet and Maple will be presented.
For a set \( S \) of two or more vertices in a nontrivial connected graph \( G \) of order \( n \), a collection \(\{T_1, T_2, \ldots, T_\ell\}\) of trees in \( G \) is said to be an internally disjoint set of trees connecting \( S \) if these trees are pairwise edge-disjoint and \( V(T_i) \cap V(T_j) = S \) for every pair \( i, j \) of distinct integers with \( 1 \leq i, j \leq \ell \). For an integer \( k \) with \( 2 \leq k \leq n \), the tree \( k \)-connectivity \( \kappa_k(G) \) of \( G \) is the greatest positive integer \( \ell \) for which \( G \) contains at least \( \ell \) internally disjoint trees connecting \( S \) for every set \( S \) of \( k \) vertices of \( G \). It is shown for every two integers \( k \) and \( r \) with \( 3 \leq k \leq 2r \) that
\[
\kappa_k(K_{r,r}) = r – \left\lceil \frac{k-1}{4} \right\rceil.
\]
This paper investigates the existence of monadic balanced ternary designs (BTDs). A monadic BTD is a BTD where each size \( K \) block contains one element that appears doubly and \( K-2 \) elements that appear singly. The authors show that the conditions
are sufficient for the existence of monadic BTDs \( (V; B; \rho_1, \rho_2, R; 4; \Lambda) \). The authors also give necessary and sufficient conditions for the existence of monadic BTDs where the block size is five and \( \Lambda \) is 3 or 6.
We consider the placement of detection devices at the vertices of a graph \( G \), where a detection device at vertex \( v \) has three possible outputs: there is an intruder at \( v \); there is an intruder at one of the vertices in the open neighborhood \( N(v) \), the set of vertices adjacent to \( v \), but which vertex in \( N(v) \) cannot be determined; or there is no intruder in \( N[v] = N(v) \cup \{v\} \). We introduce the \( 1 \)-step locating-dominating problem of placing the minimum possible number of such detection devices in \( V(G) \) so that the presence of an intruder in \( V(G) \) can be detected, and the exact location of the intruder can be identified, either immediately or when the intruder has moved to an adjacent vertex. Some related problems are introduced.
Recently, four new vertex colorings of graphs (in which adjacent vertices may be colored the same) were introduced for the purpose of distinguishing every pair of adjacent vertices. For each graph and for each of these four colorings, the minimum number of required colors never exceeds the chromatic number of the graph. In this paper, we summarize some of the results obtained on these colorings and introduce some relationships among them.
We address the problem: for which values of \( d \) and \( n \) does there exist a triangle-free regular graph of degree \( d \) on \( n \) vertices? A complete solution is given.
Let \( G = K_{a,b} \), where \( a, b \) are even, or \( G = K_{a,a} – M_{2a} \), where \( a \geq 1 \) is an odd integer and \( M_{2a} \) is a perfect matching in \( K_{a,a} \). It has been shown ([3,4]) that \( G \) is arbitrarily decomposable into closed trails. Billington asked if the graph \( K_{r,s} – F \), where \( s, r \) are odd and \( F \) is a (smallest possible) spanning subgraph of odd degree, is arbitrarily decomposable into closed trails ([2]).
In this article we answer the question in the affirmative.
This paper considers the Lehmer matrix and its recursive analogue. The determinant of the Lehmer matrix is derived explicitly by both its LU and Cholesky factorizations. We further define a generalized Lehmer matrix with \((i,j)\) entries \( g_{ij} = \frac{\text{min} \{u_{i+1}, u_{j+1}\}}{\text{max} \{u_{i+1}, u_{j+1}\}} \) where \( u_n \) is the \( n \)th term of a binary sequence \(\{u_n\}\). We derive both the LU and Cholesky factorizations of this analogous matrix and we precisely compute the determinant.
Random number generators are a small part of any computer simulation project. Yet they are the heart and the engine that drives the project. Often times software houses fail to understand the complexity involved in building a random number generator that will satisfy the project requirements and will be able to produce realistic results. Building a random number generator with a desirable periodicity, that is uniform, that produces all the random permutations with equal probability, and at random, is not an easy task. In this paper we provide tests and metrics for testing random number generators for uniformity and randomness. These tests are in addition to the already existing tests for uniformity and randomness, which we modify by running each test a large number of times on sub-sequences of random numbers, each of length \( n \). The test result obtained each time is used to determine the probability distribution function. This eliminates the random number generator misclassification error. We also provide new tests for uniformity and randomness, the new tests for uniformity test the skewness of each one of the subgroups as well as the kurtosis. The tests for randomness, which include the Fourier spectrum, the phase spectrum, the discrete cosine transform spectrum, and the orthogonal wavelet domain, test for patterns not detected in the raw data space. Finally we provide visual and acoustic tests.
For a connected graph \( G \) of order \( n \), the detour distance \( D(u, v) \) between two vertices \( u \) and \( v \) in \( G \) is the length of a longest \( u-v \) path in \( G \). A Hamiltonian labeling of \( G \) is a function \( c: V(G) \to \mathbb{N} \) such that
\[ |c(u) – c(v)| + D(u,v) \geq n \]
for every two distinct vertices \( u \) and \( v \) of \( G \). The value \( \text{hn}(c) \) of a Hamiltonian labeling \( c \) of \( G \) is the maximum label (functional value) assigned to a vertex of \( G \) by \( c \); while the Hamiltonian labeling number \( \text{hn}(G) \) of \( G \) is the minimum value of a Hamiltonian labeling of \( G \). We present several sharp upper and lower bounds for the Hamiltonian labeling number of a connected graph in terms of its order and other distance parameters.
A graph \( G \) is \( 3 \)-existentially closed (\( 3 \)-e.c.) if each \( 3 \)-set of vertices can be extended in all of the possible eight ways. Results which improve the lower bound of the minimum order of a \( 3 \)-e.c. graph are reported. It has been shown that \( m_{ec}(3) \geq 24 \), where \( m_{ec}(3) \) is defined to be the minimum order of a \( 3 \)-e.c. graph.
In this study, we analyze the structure of the full collineation group of certain Veblen-Wedderburn (VW) planes of orders \( 5^2 \), \( 7^2 \), and \( 11^2 \). We also discuss a reconstruction method using their collineation groups.
A Sarvate-Beam Quad System \( SB(v, 4) \) is a set \( V \) of \( v \) elements and a collection of \( 4 \)-subsets of \( V \) such that each distinct pair of elements in \( V \) occurs \( i \) times for every \( i \) in the list \( 1, 2, \ldots, \binom{v}{2} \). In this paper, we completely enumerate all Sarvate-Beam Quad Systems for \( v = 6 \).
In this paper, we present (by using Cauchy-Schwarz inequalities) some new results amongst the parameters of balanced arrays (B-arrays) with two symbols and having strength four, which are necessary for the existence of such balanced arrays. We then discuss and illustrate their use and applications.
The Oberwolfach problem (OP) asks whether \( K_n \) (for \( n \) odd) or \( K_n \) minus a \( 1 \)-factor (for \( n \) even) admits a \( 2 \)-factorization where each \( 2 \)-factor is isomorphic to a given \( 2 \)-factor \( F \). The order \( n \) and the type of the \( 2 \)-factor \( F \) are the parameters of the problem. For \( n \leq 17 \), the existence of a solution has been resolved for all possible parameters. There are also many special types of \( 2 \)-factors for which solutions to OP are known. We provide solutions to OP for all orders \( n \), \( 18 \leq n \leq 40 \). The computational results for higher orders were obtained using the SHARCNET high-performance computing cluster.
Let \( f_6(n) \) denote the number of partitions of the natural number \( n \) into parts co-prime to \( 6 \). This function was originally studied by Schur. We derive two explicit formulas for \( f_6(n) \), one of them in terms of the partition function \( p(n) \). We also derive three recurrences for \( f_6(n) \).
Chain integrator backstepping is a recursive design tool that has been used in nonlinear control systems. The complexity of the computation of the chain integrator backstepping control law makes inevitable the use of a computer algebra system. A recursive algorithm is designed to compute the integrator backstepping control process. A computer algebra program (Maple procedure) is developed for symbolic computation of the control function using a newly developed recursive algorithm. We will present some demonstrative examples to show the stability of the control systems using Lyapunov functions.
Chain integrator backstepping is a recursive design tool that has been used in nonlinear control systems. The complexity of the computation of the chain integrator backstepping control law makes inevitable the use of a computer algebra system. A recursive algorithm is designed to compute the integrator backstepping control process. A computer algebra program (Maple procedure) is developed for symbolic computation of the control function using a newly developed recursive algorithm. We will present some demonstrative examples to show the stability of the control systems using Lyapunov functions.
Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A \) be an abelian group. A labeling \( f: V(G) \to A \) induces an edge labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E(G) \). For each \( i \in A \), let \( v_f(i) = \text{card}\{v \in V(G) \mid f(v) = i\} \) and \( e_f(i) = \text{card}\{e \in E(G) \mid f^*(e) = i\} \). Let \( c(f) = \{\lvert e_f(i) – e_f(j) \rvert \mid (i, j) \in A \times A\} \). A labeling \( f \) of a graph \( G \) is said to be \( A \)-friendly if \( \lvert v_f(i) – v_f(j) \rvert \leq 1 \) for all \( (i, j) \in A \times A \). If \( c(f) \) is a \( (0, 1) \)-matrix for an \( A \)-friendly labeling \( f \), then \( f \) is said to be \( A \)-cordial. When \( A = \mathbb{Z}_2 \), the friendly index set of the graph \( G \), \( FI(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert \mid \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\} \). In \([15]\) the friendly index set of a cycle is completely determined. We consider the friendly index sets of broken wheels with three spokes.
The intractability of the traditional discrete logarithm problem (DLP) forms the basis for the design of numerous cryptographic primitives. In \([2]\) M. Sramka et al. generalize the DLP to arbitrary finite groups. One of the reasons mentioned for this generalization is P. Shor’s quantum algorithm \([4]\) which solves efficiently the traditional DLP. The DLP for a non-abelian group is based on a particular representation of the group and a choice of generators. In this paper, we show that care must be taken to ensure that the representation and generators indeed yield an intractable DLP. We show that in \(\text{PSL}(2,p) = \langle \alpha, \beta \rangle\) the generalized discrete logarithm problem with respect to \((\alpha,\beta)\) is easy to solve for a specific representation and choice of generators \(\alpha\) and \(\beta\). As a consequence, such representation of \(\text{PSL}(2,p)\) and generators should not be used to design cryptographic primitives.
Beautifully Ordered Balanced Incomplete Block Designs, \(\text{BOBIBD}(v, k, \lambda, k_1, \lambda_1)\), are defined and the proof is given to show that necessary conditions are sufficient for the existence of BOBIBDs with block size \(k = 3\) and \(k = 4\) for \(k_1 = 2\) except possibly for eleven exceptions. Existence of BOBIBDs with block size \(k = 4\) and \(k_1 = 3\) is demonstrated for all but one infinite family and the non-existence of \(\text{BOBIBD}(7, 4, 2, 3, 1)\), the first member of the unknown series, is shown.
Let \(G\) be a graph. Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) with \(g(x) \leq f(x)\) for any \(x \in V(G)\). 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 \(E_x = \{e : e = xy \in E(G)\}\). A graph \(G\) is said to be fractional \((g, f, n)\)-critical if \(G – N\) has a fractional \((g, f)\)-factor for each \(N \subseteq V(G)\) with \(|N| = n\). In this paper, several sufficient conditions in terms of stability number and degree for graphs to be fractional \((g, f, n)\)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.
The modified Zagreb indices are important topological indices in mathematical chemistry. In this paper, we study the modified Zagreb indices of disjunctions and symmetric differences.
Given a graph \(G\) and a non-negative integer \(g\), the \(g\)-extra-connectivity of \(G\) (written \(\kappa_g(G)\)) is the minimum cardinality of a set of vertices of \(G\), if any, whose deletion disconnects \(G\), and every remaining component has more than \(g\) vertices. The usual connectivity and superconnectivity of \(G\) correspond to \(\kappa_0(G)\) and \(\kappa_1(G)\), respectively. In this paper, we determine \(\kappa_g(P_{n_1} \times P_{n_2} \times \cdots \times P_{n_s})\) for \(0 \leq g \leq s\), where \(\times\) denotes the Cartesian product of graphs. We generalize \(\kappa_g(Q_n)\) for \(0 \leq g \leq n\), \(n \geq 4\), where \(Q_n\) denotes the \(n\)-cube.
A graph labeling is an assignment of integers (labels) to the vertices and/or edges of a graph. Within vertex labelings, two main branches can be distinguished: difference vertex labelings that associate each edge of the graph with the difference of the labels of its endpoints. Graceful and edge-antimagic vertex labelings correspond to these branches, respectively. In this paper, we study some connections between them. Indeed, we study the conditions that allow us to transform any \(a\)-labeling (a special case of graceful labeling) of a tree into an \((a, 1)\)- and \((a, 2)\)-edge antimagic vertex labeling.
The domination number \(\gamma(G)\) of a graph \(G\) is the minimum cardinality among all dominating sets of \(G\), and the independence number \(\alpha(G)\) of \(G\) is the maximum cardinality among all independent sets of \(G\). For any graph \(G\), it is easy to see that \(\gamma(G) \leq \alpha(G)\). In this paper, we present a characterization of trees \(T\) with \(\gamma(T) = \alpha(T)\).
This paper generalizes the concept of locally connected graphs. A graph \(G\) is triangularly connected if for every pair of edges \(e_1, e_2 \in E(G)\), \(G\) has a sequence of \(3\)-cycles \(C_1, C_2, \ldots, C_l\) such that \(e_1 \in C_1, e_2 \in C_l\) and \(E(C_i) \cap E(C_{i+1}) \neq \emptyset\) for \(1 \leq i \leq l-1\). In this paper, we show that every triangularly connected \(K_{1,4}\)-free almost claw-free graph on at least three vertices is fully cycle extendable.
Let \(G = (V,E)\) be a simple graph. \({N}\) and \({Z}\) denote the set of all positive integers and the set of all integers, respectively. The sum graph \(G^+(S)\) of a finite subset \(S \subset{N}\) is the graph \((S, {E})\) with \(uv \in {E}\) if and only if \(u+v \in S\). \(G\) is a sum graph if it is isomorphic to the sum graph of some \(S \subseteq {N}\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices, which result in a sum graph when added to \(G\). By extending \({N}\) to \({Z}\), the notions of the integral sum graph and the integral sum number of \(G\) are obtained, respectively. In this paper, we prove that \(\zeta(\overline{C_n}) = \sigma(\overline{C_n}) = 2n-7\) and that \(\zeta(\overline{W_n}) = \sigma(\overline{W_n}) = 2n-8\) for \(n \geq 7\).
We investigate the relationship between geodetic sets, \(k\)-geodetic sets, dominating sets, and independent sets in arbitrary graphs. As a consequence of the study, we provide several tight bounds on the geodetic number of a graph.
For \(1 \leq d \leq v-1\), let \(V\) denote the \(2v\)-dimensional symplectic space over a finite field \({F}_q\), and fix a \((v-d)\)-dimensional totally isotropic subspace \(W\) of \(V\). Let \({L}(d, 2v) = {P}\cup \{V\}\), where \({P} = \{A \mid A \text{ is a subspace of } V, A \cap W = \{0\} \text{ and } A \subset W^\perp\}\). Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.
Let \(M\) be a graph, and let \(H(M)\) denote the homeomorphism class of \(M\), that is, the set of all graphs obtained from \(M\) by replacing every edge by a `chain’ of edges in series. Given \(M\) it is possible, either using the `chain polynomial’ introduced by E. G. Whitehead and myself (Discrete Math. \(204(1999) 337-356)\) or by ad hoc methods, to obtain an expression which subsumes the chromatic polynomials of all the graphs in \(H(M)\). It is a function of the number of colors and the lengths of the chains replacing the edges of \(M\). This function contains complete information about the chromatic properties of these graphs. In particular, it holds the answer to the question “Which pairs of graphs in \(H(M)\) are chromatically equivalent”. However, extracting this information is not an easy task.
In this paper, I present a method for answering this question. Although at first sight it appears to be wildly impractical, it can be persuaded to yield results for some small graphs. Specific results are given, as well as some general theorems. Among the latter is the theorem that, for any given integer \(\gamma\), almost all cyclically \(3\)-connected graphs with cyclomatic number \(\gamma\) are chromatically unique.
The analogous problem for the Tutte polynomial is also discussed, and some results are given.
Let \(G\) be a simple graph of order \(p \geq 2\). A proper \(k\)-total coloring of a simple graph \(G\) is called a \(k\)-vertex distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The notation \(\chi_{vt}(G)\) indicates the smallest number of colors required for which \(G\) admits a \(k\)-VDTC with \(k \geq \chi_{vt}(G)\). For every integer \(m \geq 3\), we will present a graph \(G\) of maximum degree \(m\) such that \(\chi_{vt}(G) < \chi_{vt}(H)\) for some proper subgraph \(H \subseteq G\).
Let \(G = (V,E)\) be a graph. Let \(\gamma(G)\) and \(\gamma_t(G)\) be the domination and total domination number of a graph \(G\), respectively. The \(\gamma\)-criticality and \(\gamma_t\)-criticality of Harary graphs are studied. The Question \(2\) of the paper [W. Goddard et al., The Diameter of total domination vertex critical graphs, Discrete Math. \(286 (2004), 255-261]\) is fully answered with the family of Harary graphs. It is answered to the second part of Question \(1\) of that paper with some Harary graphs.
Let \(G\) be a connected graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2} \sum_{u,v \in V(G)} d^2(u,v),\) with the summation going over all pairs of vertices in \(G\) and \(d(u,v)\) denotes the distance between \(u\) and \(v\) in \(G\). In this paper, we determine the upper or lower bounds on hyper-Wiener index of trees with given number of pendent vertices, matching number, independence number, domination number, diameter, radius, and maximum degree.
A large set of resolvable Mendelsohn triple systems of order \(v\), denoted by \(\text{LRMTS}(v)\), is a collection of \(v-2\) \(\text{RMTS}(v)\)s based on \(v\)-set \(X\), such that every Mendelsohn triple of \(X\) occurs as a block in exactly one of the \(v-2\) \(\text{RMTS}(v)\)s. In this paper, we use \(\text{TRIQ}\) and \(\text{LR-design}\) to present a new product construction for \(\text{LRMTS}(v)\)s. This provides some new infinite families of \(\text{LRMTS}(v)\)s.
In this paper, we investigate the existence of nontrivial solutions for the equation \(y(G \Box H) – \gamma(G) \gamma(H)\) fixing one factor. For the complete bipartite graphs \(K_{m,n}\), we characterize all nontrivial solutions when \(m = 2, n \geq 3\) and prove the nonexistence of solutions when \(m \geq 2, n \leq 3\). In addition, it is proved that the above equation has no nontrivial solution if \(A\) is one of the graphs obtained from \(G\), the cycle of length \(n\), either by adding a vertex and one pendant edge joining this vertex to any vertex to any \(v\in V(C_n)\), or by adding one chord joining two alternating vertices of \(C_n\).
For a graph \(G = (V(G), E(G))\), let \(i(G)\) be the number of isolated vertices in \(G\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\) if \(G\) is not complete; \(I(G) = |V(G)|-1\) otherwise. In this paper, several sufficient conditions in terms of isolated toughness are obtained for the existence of \([a, b]\)-factors avoiding given subgraphs, e.g., a set of vertices, a set of edges and a matching, respectively.
In a graph \(G\), the distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining them. The eccentricity \(e(u)\) of a vertex \(u\) is the distance to a vertex farthest from \(u\). The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph \(R(G)\) based on \(G\) has the vertex set as in \(G\). Two vertices \(u\) and \(v\) are adjacent in \(R(G)\) if the distance between them in \(G\) is equal to the radius of \(G\). If \(G\) is disconnected, then two vertices are adjacent in \(R(G)\) if they belong to different components. The main objective of this paper is to find a necessary and sufficient condition for a graph to be a radial graph.
Let \(\{T, T’\}\) be a Latin bitrade. Then \(T\) (and \(T’\)) is said to be \((r,c,e)\)-homogeneous if each row contains precisely \(r\) entries, each column contains precisely \(c\) entries, and each entry occurs precisely \(e\) times. An \((r,c,e)\)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely \((r,c,e) = (3,3,3), (4,4,2)\), or \((6,3,2)\). The first case has been completely classified by a number of authors. We present classifications for the other two cases.
In this paper, we prove an interesting property of rook polynomials for \(2\)-D square boards and extend that for rook polynomials for \(3\)-D cubic, and \(r\)-D “hypercubic” boards. In particular, we prove that for \(r\)-D rook polynomials the modulus of the sum of their roots equals their degree. We end with some further questions, mainly for the \(2\)-D and \(3\)-D case, that could serve as future projects.
Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\), then the subgraph \(H\) 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 \(\lambda V(G)\) 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 in \(G\) 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 the large set of \(\lambda\)-fold \(P_3\)-factors in \(K_{v,v}\) and obtain its existence spectrum.
Let \(k \geq 1\), \(l \geq 3\), and \(s \geq 5\) be integers. In \(1990\), Erdős and Faudree conjectured that if \(G\) is a graph of order \(4k\) with \(\delta(G) \geq 2k\), then \(G\) contains \(k\) vertex-disjoint \(4\)-cycles. In this paper, we consider an analogous question for \(5\)-cycles; that is to say, if \(G\) is a graph of order \(5k\) with \(\delta(G) \geq 3k\), then \(G\) contains \(k\) vertex-disjoint \(5\)-cycles? In support of this question, we prove that if \(G\) is a graph of order \(5k\) with \(\omega_2(G) \geq 6l – 2\), then, unless \(\overline{K_{l-2}} + K_{2l+1,2l+1} \subseteq G \subseteq K_{l-2} + K_{2l+1,2l+1}\), \(G\) contains \(l – 1\) vertex-disjoint \(5\)-cycles and a path of order \(5\), which is vertex-disjoint from the \(l – 1\) \(5\)-cycles. In fact, we prove a more general result that if \(G\) is a graph of order \(5k + 2s\) with \(\omega_2(G) \geq 6k + 2s\), then, unless \(\overline{K_{k}} + K_{2k+s,2k+s} \subseteq G \subseteq K_{k} + K_{2k+s,2k+s}\), \(G\) contains \(k+1\) vertex-disjoint \(5\)-cycles and a path of order \(2s – 5\), which is vertex-disjoint from the \(k + 1\) \(5\)-cycles. As an application of this theorem, we give a short proof for determining the exact value of \(\text{ex}(n,(k + 1)C_5)\), and characterize the extremal graph.
In this paper, we present the complex factorizations of the Jacobsthal and Jacobsthal Lucas numbers by determinants of tridiagonal matrices.
In this paper, we find families of \((0, -1, 1)\)-tridiagonal matrices whose determinants and permanents equal the negatively subscripted Fibonacci and Lucas numbers. Also, we give complex factorizations of these numbers by the first and second kinds of Chebyshev polynomials.
We classify all finite near hexagons which satisfy the following properties for a certain \(t_2 \in \{1,2,4\}\):(i) every line is incident with precisely three points;(ii) for every point \(x\), there exists a point \(y\) at distance \(3\) from \(x\);(iii) every two points at distance \(2\) from each other have either \(1\) or \(t_2 + 1\) common neighbours;(iv) every quad is big. As a corollary, we obtain a classification of all finite near hexagons satisfying (i), (ii) and (iii) with \(t_2\) equal to \(4\).
In this paper, we obtain the largest Laplacian spectral radius for bipartite graphs with given matching number and use them to characterize the extremal general graphs.
For integers \(k, \theta \leq 3\) and \(\beta \geq 1\), an integer \(k\)-set \(S\) with the smallest element \(0\) is a \((k; \beta, \theta)\)-free set if it does not contain distinct elements \(a_{i,j}\) (\(1 \leq i \leq j \leq \theta\)) such that \(\sum_{j=1}^{\theta -1}a_{i ,j} = \beta a_{i_\theta}\). The largest integer of \(S\) is denoted by \(\max(S)\). The generalized antiaverage number \(\lambda(k; \beta, \theta)\) is equal to \(\min\{\max(S) : S \text{ is a } (k^0; \delta, 0)\text{-free set}\}\). We obtain:(1) If \(\beta \notin \{\theta-2, \theta-1, \theta\}\), then \(\lambda(m; \beta, \theta) \leq (\theta-1)(m-2) + 1\); (2) If \(\beta \geq {\theta-1}\), then \(\lambda(k; \beta, \theta) \leq \min\limits_{k=m+n}\{\lambda(m;\beta,\theta)+\beta \lambda (n;\beta,\theta)+1\}\), where \(k =m+n \) with \(n>m\geq 3\) and \(\lambda(2n;\beta,\theta)\leq \lambda(n;\beta,\theta)(\beta+1)+\varepsilon\), for \(\varepsilon=1\) for \(\theta=3\) and \(\varepsilon=0\) otherwise.
A connected graph is highly irregular if the neighbors of each vertex have distinct degrees. We will show that every highly irregular tree has at most one nontrivial automorphism. The question that motivated this work concerns the proportion of highly irregular trees that are asymmetric, i.e., have no nontrivial automorphisms. A \(d\)-tree is a tree in which every vertex has degree at most \(d\). A technique for enumerating unlabeled highly irregular \(d\)-trees by automorphism group will be described for \(d \geq 4\) and results will be given for \(d = 4\). It will be shown that, for fixed \(d\), \(d \geq 4\), almost all highly irregular \(d\)-trees are asymmetric.
Combining with specific degrees or edges of a graph, this paper provides some new classes of upper embeddable graphs and extends the results in [Y. Huang, Y. Liu, Some classes of upper embeddable graphs, Acta Mathematica Scientia, \(1997, 17\)(Supp.): \(154-161\)].
A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral trees \(S(r;m_i) = S(a_1+a_2+\cdots+a_s;m_1,m_2,\ldots,m_s)\) of diameter \(4\) with \(s = 2,3\). We give a better sufficient and necessary condition for the tree \(S(a_1+a_2;m_1,m_2)\) of diameter \(4\) to be integral, from which we construct infinitely many new classes of such integral trees by solving some certain Diophantine equations. These results are different from those in the existing literature. We also construct new integral trees \(S(a_1+a_2+a_3;m_1,m_2,m_3) = S(a_1+1+1;m_1,m_2,m_3)\) of diameter \(4\) with non-square numbers \(m_2\) and \(m_3\). These results generalize some well-known results of P.Z. Yuan, D.L. Zhang \(et\) \(al\).
Zagreb indices are the best known topological indices which reflect certain structural features of organic molecules. In this paper we point out that the modified Zagreb indices are worth studying and present some results about product graphs.
Let \(g \in H(\mathcal{B})\), \(g(0) = 0\) and \(\varphi\) be a holomorphic self-map of the unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). The following integral-type operator
\[I_\varphi^g(f)(z) = \int_{0}^{1} {\mathcal{R}f(\varphi(tz))}{g(tz)}\frac{ dt}{t}, \quad f \in H(\mathbb{B}),z\in \mathbb{B},\]
was recently introduced by S. Stević and studied on some spaces of holomorphic functions on \(\mathbb{B}\), where \(\mathcal{R}f(z) = \sum_{k=1}^n z_k \frac{\partial f}{\partial z_k}(z)\) is the radial derivative of \(g\). The boundedness and compactness of this operator from generally weighted Bloch spaces to Bloch-type spaces on \(\mathbb{B}\) are investigated in this note.
We start by proving that the Henson graphs \(H_n\), \(n \geq 3\) (the homogeneous countable graphs universal for the class of all finite graphs omitting the clique of size \(n\)), are retract rigid. On the other hand, we provide a full characterization of retracts of the complement of \(H_3\). Further, we prove that each countable partial order embeds in the natural order of retractions of the complements of Henson graphs. Finally, we show that graphs omitting sufficiently large null subgraphs omit certain configurations in their endomorphism monoids.
Combining integration method with series rearrangement,we establish several closed formulae for Gauss hypergeometric series with four free parameters, which extend essentially the related results found recently by Elsner \((2005).\)
In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation
\(x_{n+1} = \frac{Ax_{n-2l}}{B+C \prod\limits_{i=0}^{2k}x_{n-i}}, n=0,1,\ldots ,\)
where \(A\), \(B\), \(C\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\), \(l\) are nonnegative integers, \(l \leq k\). Also, we derive solutions of some special cases of this equation.
In this paper, the critical group structure of the Cartesian product graph \(C_4 \times C_n\) is determined, where \(n \geq 3\).
Let \(G = (V, E)\) be a simple connected graph with \(7\) vertices. The degree of \(v_i \in V\) and the average of degrees of the vertices adjacent to \(v_i\) are denoted by \(d_i\) and \(m_i\), respectively. The spectral radius of \(G\) is denoted by \(\rho(G)\). In this paper, we introduce a parameter into an equation of adjacency matrix, and obtain two inequalities for upper and lower bounds of spectral radius. By assigning different values to this parameter, one can obtain some new and existing results on spectral radius. Specially, if \(G\) is a nonregular graph, then
\[\rho(G) \leq \max_{1 \leq j<i \leq n} \{ \frac{d_i m_i – d_j m_j + \sqrt{(d_i m_i – d_j m_j)^2 – 4d_i d_j(d_i-d_j) (m_i – m_j)}}{2(d_i-d_j)} \}\] and \[\rho(G)\geq \min_{1 \leq j<i \leq n} \{ \frac{d_i m_i – d_j m_j + \sqrt{(d_i m_i – d_j m_j)^2 – 4d_i d_j(d_i-d_j) (m_i – m_j)}}{2(d_i-d_j)} \}.\] If \(G\) is a bidegreed graph whose vertices of same degree have equal average of degrees, then the equality holds.
An orientation of a simple graph \(G\) is called an oriented graph. If \(D\) is an oriented graph, \(\delta(D)\) its minimum degree and \(\lambda(D)\) its edge-connectivity, then \(\lambda(D) \leq \delta(D)\). The oriented graph is called maximally edge-connected if \(\lambda(D) = \delta(D)\) and super-edge-connected, if every minimum edge-cut is trivial. In this paper, we show that an oriented graph \(D\) of order \(n\) without any clique of order \(p + 1\) in its underlying graph is maximally edge-connected when
\[n \leq 4{\lfloor\frac{p\delta(D)}{p – 1}\rfloor}-1.\]
Some related conditions for oriented graphs to be super-edge-connected are also presented.
Denote by \(\mathcal{A_n}\), the set of the polyphenyl chains with \(n\) hexagons. For any \(A_n \in \mathcal{A_n}\), let \(m_k(A_n)\) and \(i_k(A_n)\) be the numbers of \(k\)-matchings and \(k\)-independent sets of \(A_n\), respectively. In the paper, we show that for any \(A_n \in \mathcal{A_n}\) and for any \(k \geq 0\),\(m_k(M_n) \leq m_k(A_n) \leq m_k(O_n) \quad \text{and} \quad i_k(M_n) \geq i_k(A_n) \geq i_k(O_n),\) with the equalities holding if \(A_n = M_n\) or \(A_n = O_n\), where \(M_n\) and \(O_n\) are the meta-chain and the ortho-chain, respectively. These generalize some related results in \([1]\).
Let \(G = (X, Y, E(G))\) be a bipartite graph with vertex set \(V(G) = X ! Y\) and edge set \(E(G)\), and let \(g, f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\); a \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \(\mathcal{F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly \(r\) edges in common with \(H\), we say that \(F_i\) is \(r\)-orthogonal to \(H\). In this paper, it is proved that every bipartite \((0, mf-(m-1)r)\)-graph has \((0, f)\)-factorizations randomly \(r\)-orthogonal to any given subgraph with \(m\) edges if \(2r \leq f(x)\) for any \(x \in V(G)\).
We define an \(r\)-capacitated dominating set of a graph \(G = (V,E)\) as a set \(\{v_1, \ldots, v_k\} \subseteq V\) such that there is a partition \((V_1, \ldots, V_k)\) of \(V\) where for all \(i\), \( v_i \in V_i\), \(v_i\) is adjacent to all of \(V_i – \{v_i\}\), and \(|V_i| \leq r + 1\). \(\daleth_r(G)\) is the minimum cardinality of an \(r\)-capacitated dominating set. We show properties of \(\daleth_r\), especially as regards the trivial lower bound \(|V|/(r + 1)\). We calculate the value of the parameter in several graph families, and show that it is related to codes and polyominoes. The parameter is \(NP\)-complete in general to compute, but a greedy approach provides a linear-time algorithm for trees.
On the basis of joint trees introduced by Yanpei Liu, by choosing different spanning trees and classifying the associated surfaces, we obtain the explicit expressions of genus polynomials for three types of graphs, namely \(K_5^n, W_6^n\) and \(K_{3,3}^n\), which are different from the graphs whose embedding distributions by genus have been obtained. And \(K_5^n\) and \(K_{3,3}^n\) are non-planar.
We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.
A \((d,1)\)-totel labelling of a graph \(G\) is an assignment of integers to \(V(G) \cap E(G)\) such that: (i) any two adjacent vertices of \(G\) receive distinct integers, (ii) any two adjacent edges of \(G\) receive distinct integers, and (iii) a vertex and its incident edge receive integers that differ by at least \(d\) in absolute value. The span of a \((d,1)\)-total labelling is the maximum difference between two labels. The minimum span of labels required for such a \((d, 1)\)-total labelling of \(G\) is called the \((d, 1)\)-total number and is denoted by \(\lambda_d^T(G)\). In this paper, we prove that \(\lambda_d^T(G)\geq d+r+1 \) for \(r\)-regular nonbipartite graphs with \(d \geq r \geq 3\) and determine the \((d, 1)\)-total numbers of flower snarks and of quasi flower snarks.
Let \(G = (V,E)\) be a simple graph with the vertex set \(V\) and the edge set \(E\). \(G\) is a sum graph if there exists a labelling \(f\) of the vertices of \(G\) into distinct positive integers such that \(uv \in E\) if and only if \( f(w)=f(u) + f(v) \) for some vertex \(w \in V\). Such a labelling \(f\) is called a sum labelling of \(G\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which result in a sum graph when added to \(G\). Similarly, the integral sum graph and the integral sum number \(\zeta(G)\) are also defined. The difference is that the labels may be any distinct integers.
In this paper, we will determine that
\[\begin{cases}
0 = \zeta(\overline{P_4}) < \sigma(\overline{P_4}) = 1;\\
1 = \zeta(\overline{P_5}) < \sigma(\overline{P_5}) = 2;\\
3 = \zeta(\overline{P_6}) < \sigma(\overline{P_6}) = 4;\\
\zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 0, \text{ for } n = 1, 2, 3;\\
\zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 2n – 7, \text{ for } n \geq 7;
\end{cases}\]
and
\[\begin{cases}
0 = \zeta(\overline{F_5}) < \sigma(\overline{F_5}) = 1;\\
2 = \zeta(\overline{F_5}) < \sigma(\overline{F_6}) = 2;\\
\zeta(\overline{F_c}) = \sigma(\overline{F_n}) = 0, \text{ for } n =3,4;\\
\zeta(\overline{F_n}) = \sigma(\overline{F_n}) = 2n – 8, \text{ for } n \geq 7.
\end{cases}\]
The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI indices of bicyclic graphs whose cycles do not share two or more common vertices.
For a graph \( G = (V, E) \), a non-empty set \( S \subseteq V \) is a global offensive alliance (respectively, global strong offensive alliance) if for every vertex \( v \in V – S \), at least half of the vertices in its closed neighborhood are in \( S \) (respectively, a strict majority of its closed neighborhood are in \( S \)). The global offensive alliance number \( \gamma_o(G) \) (respectively, global strong offensive alliance number \( \gamma_{\hat{o}}(G) \)) is the minimum cardinality of a global offensive alliance (respectively, global strong offensive alliance) of \( G \). In this paper, we determine an upper bound on each parameter for bipartite graphs without isolated vertices. More precisely, we show that \( \gamma_o(G) \leq \frac{n – \ell + s}{2} \) and \( \gamma_{\hat{o}}(G) \leq \frac{n + \ell}{2} \), where \( n \), \( \ell \), and \( s \) are the order, the number of leaves, and the support vertices of \( G \), respectively. Moreover, extremal trees attaining each bound are characterized.
There is a hypothesis that a non-self-centric radially-maximal graph of radius \( r \) has at least \( 3r – 1 \) vertices. Moreover, if it has exactly \( 3r – 1 \) vertices, then it is planar with minimum degree \( 1 \) and maximum degree \( 3 \). Using an enhanced exhaustive computer search, we prove this hypothesis for \( r = 4, 5 \).
A vertex set \( X \) of a simple graph is called OO-irredundant if for each \( v \in X \), \( N(v) – N(X – \{v\}) \neq \emptyset \). Basic results for maximal OO-irredundant sets of a graph are obtained.
A set of necessary conditions for the existence of a partition of \(\{1, \ldots, 2m – 1, L\}\) into differences \(d, d + 1, \ldots, d + m – 1\) is \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\) and \(m \geq 2d – 2\) or \(m = 1\). If \(m = 2d – 2\) then \(L = 5d – 5\), if \(m = 2d – 1\) then \(4d – 2 \leq L \leq 6d – 4\) and if \(m \geq 2d\) then \(2m \leq L \leq 3m + d – 2\). Similar conditions for the partition of \(\{1, \ldots, 2m, L\} \setminus \{2\}\) into differences \(d, d + 1, \ldots, d + m – 1\) are \((m, L) \equiv (0, 0), (1, d + 1), (2, 1), (3, d) \pmod{(4, 2)}\), \((d, m, L) \neq (1, 1, 4), (2, 3, 8)\) and \(m \geq 2d – 2\), \(m = 1\) or \((d, m, L) = (3, 2, 7), (3, 3, 9)\). If \(m = 2d – 2\) then \(L = 5d – 5, 5d – 3\), if \(m = 2d – 1\) then \(4d – 1 \leq L \leq 6d – 2\) and if \(m \geq 2d\) then \(2m + 1 \leq L \leq 3m + d – 1\).
It is shown that for many cases when the necessary conditions hold, the set \(\{1, \ldots, 2m – 1, L\}\) and \(\{1, \ldots, 2m – 1, L\} \setminus \{2\}\) can be so partitioned. These partitions exist for all the minimum and maximum \(L\), when \(d \leq 3\), when \(m = 1\) and when \(m \geq 8d – 3\) (\(m \geq 8d + 4\) in the hooked case). The constructions given fully solve the existence of these partitions if the necessary conditions for the existence of extended and hooked extended Langford sequences are sufficient.
Let \( n \) and \( q \) be positive integers, with \( n \geq 3 \), and let \( N = nq \). As input, we are to be given an arbitrary ordered \( n \)-sequence \( x_1, x_2, \ldots, x_n \), where \( 1 \leq x_i \leq N \) for all \( i \). We are to be presented with this sequence one entry at a time. As each entry is received, it must be placed into one of the positions \( 1, 2, \ldots, n \), where it must remain. A natural way to do this, in an attempt to sort the input sequence, is as follows. For any integer \( x \in \{1, \ldots, N\} \), we let \( s(x) \) denote the unique integer \( s \) for which \( (s-1)q + 1 \leq x \leq sq \). When we receive the entry \( x_i \), we consider those positions still unoccupied after having placed the previous \( i-1 \) entries, and we place \( x_i \) into the one which is closest to \( s(x_i) \). In the event of a tie for closest, we choose the higher of the two positions. We refer to this procedure as the \({placement\; algorithm}\). Regarding this algorithm, we consider the following question: for how many input sequences will the last two positions filled be positions \( 1 \) and \( n \)? We show that this number is \( (n-1)^{n-3}n^2q^n \).
The complexity of the maximum leaf spanning tree problem for grid graphs is currently unknown. We determine the maximum number of leaves in a grid graph with up to \(4\) rows and with \(6\) rows. Furthermore, we give some constructions of spanning trees of grid graphs with a large number of leaves.
Let \((X, \mathcal{B})\) be a \(\lambda\)-fold \(G\)-decomposition of \(\lambda H\). Let \(G_i\), \(i=1,\ldots,\mu\), be all nonisomorphic proper subgraphs of \(G\) without isolated vertices. Put \(\mathcal{B}_i = \{B_i \mid B \in \mathcal{B}\}\), where \(B_i\) is a subgraph of \(B\) isomorphic to \(G_i\). A complete simultaneous metamorphosis of \((X, \mathcal{B})\) is a rearrangement, for each \(i = 1, \ldots, \mu\), of the edges of \(\bigcup_{B \in \mathcal{B}} (E(B) \setminus E(B_i))\) 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 a \(\lambda\)-fold kite system having a complete simultaneous metamorphosis.
Whist tournaments for \( v \) players, \( \mathrm{Wh}(v) \), are known to exist for all \( v \equiv 0, 1 \pmod{4} \). In this paper, a new specialization of whist tournament design, namely a \({balanced\; whist\; tournament}\), is introduced. We establish that balanced whist tournaments on \( v \) players, \( \mathrm{BWh}(v) \), exist for several infinite classes of \( v \). An adaptation of a classic construction due to R. C. Bose and J. M. Cameron enables us to establish that \( \mathrm{BWh}(4n + 1) \) exist whenever \( 4n + 1 \) is a prime or a prime power. It is also established that \( \mathrm{BWh}(4n) \) exist for \( 4n = 2^k \), with \( k \equiv 0 \pmod{2, 3 \text{ or } 5} \). We demonstrate that a \( \mathrm{BWh}(4n + 1) \) is equivalent to a conference matrix of order \( 4n + 2 \). Consequently, a necessary condition for the existence of a \( \mathrm{BWh}(4n + 1) \) is that \( 4n + 1 \) is a product of primes each of which is \( \equiv 1 \pmod{4} \). Thus, in particular, \( \mathrm{BWh}(21) \) and \( \mathrm{BWh}(33) \) do not exist. Specific examples of \( \mathrm{BWh}(v) \) are given for \( v = 4, 8, 9, 20, 24, 32 \). It is also shown that a \( \mathrm{BWh}(12) \) does not exist.
Let \(\alpha(G)\) represent the maximal size of any product-free subset of a finite abelian group \(G\). It is well known that \(\alpha(G) = \frac{|G|}{3}\left(1 + \frac{1}{p}\right)\) if \(|G|\) is divisible by a prime \(p \equiv 2 \pmod{3}\) and \(p\) is the smallest such prime, \(\alpha(G) = \frac{|G|}{3}\) if \(|G|\) is not divisible by a prime \(p \equiv 2 \pmod{3}\) but \(3\) divides \(|G|\), and \(\alpha(G) = \frac{|G|}{3}\left(1 – \frac{1}{m}\right)\) if \(|G|\) is divisible only by primes \(p \equiv 1 \pmod{3}\) and \(m\) is the exponent of \(|G|\). In this paper, we use only basic group theory and number theory to derive exact expressions for the number of different maximal product-free subsets of \(G\) in the first two cases. The formulas are given in terms of the sizes of the subgroups of \(G\).
The \( P_3 \) intersection graph \( P_3(G) \) of a graph \( G \) is the intersection graph of all induced \( 3 \)-paths in \( G \). In this paper, we prove that any \( P_3 \)-convergent graph is \( P_3^n(G) \)-complete for some \( n \geq 1 \). Additionally, we prove that there are no \( P_3 \)-fixed graphs. The touching number, periodicity, and connectivity of \( P_3(G) \) are also studied.
This paper describes the study of a special class of 4-regular plane graphs that are Hamiltonian. These graphs are of particular interest in knot theory. An algorithm is presented that randomly generates such graphs with \( n \) vertices with a fixed (and oriented) Hamiltonian cycle in \( O(n) \) time. An exact count of the number of such graphs with \( n \) vertices is obtained, and the asymptotic growth rate of this number is determined. Numerical evidence is provided to show that the algorithm can be modified to generate these graphs with a near uniform probability. This can be considered a first step in generating large random knots without bias.
We enumerate nonisomorphic minimum genus orientable embeddings of the complete bipartite graph \( K_{m,n} \) for \( 2 \leq m, n \leq 7 \) except for \( (m, n) = (7, 7) \).
A complete coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two distinct colors \( i \) and \( j \) used in the coloring, there exist adjacent vertices of \( G \) colored \( i \) and \( j \). The maximum positive integer \( k \) for which \( G \) has a complete \( k \)-coloring is the achromatic number \( \psi(G) \) of \( G \).
A Grundy coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two colors (positive integers) \( i \) and \( j \) with \( i < j \), every vertex colored \( j \) has a neighbor colored \( i \). The maximum positive integer \( k \) for which a graph \( G \) has a Grundy \( k \)-coloring is the Grundy number \( \Gamma(G) \). Thus, \( 2 \leq \chi(G) \leq \Gamma(G) \leq \psi(G) \) for every nonempty graph \( G \). It is shown that if \( a, b, \) and \( c \) are integers with \( 2 \leq a \leq b \leq c \), then there exists a connected graph \( G \) with \( \chi(G) = a \), \( \Gamma(G) = b \), and \( \psi(G) = c \) if and only if \( a = b = c = 2 \) or \( b \geq 3 \).
A complete coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two distinct colors \( i \) and \( j \) used in the coloring, there exist adjacent vertices of \( G \) colored \( i \) and \( j \). The maximum positive integer \( k \) for which \( G \) has a complete \( k \)-coloring is the achromatic number \( \psi(G) \) of \( G \).
A Grundy coloring of a graph \( G \) is a proper vertex coloring of \( G \) with the property that for every two colors (positive integers) \( i \) and \( j \) with \( i < j \), every vertex colored \( j \) has a neighbor colored \( i \). The maximum positive integer \( k \) for which a graph \( G \) has a Grundy \( k \)-coloring is the Grundy number \( \Gamma(G) \). Thus, \( 2 \leq \chi(G) \leq \Gamma(G) \leq \psi(G) \) for every nonempty graph \( G \). It is shown that if \( a, b, \) and \( c \) are integers with \( 2 \leq a \leq b \leq c \), then there exists a connected graph \( G \) with \( \chi(G) = a \), \( \Gamma(G) = b \), and \( \psi(G) = c \) if and only if \( a = b = c = 2 \) or \( b \geq 3 \).
Let \( G \) be a simple graph, and let \( p \) be a positive integer. A subset \( D \subseteq V(G) \) is a \( p \)-dominating set of the graph \( G \) if every vertex \( v \in V(G) – D \) is adjacent to at least \( p \) vertices of \( D \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality among the \( p \)-dominating sets of \( G \). Note that the \( 1 \)-domination number \( \gamma_1(G) \) is the usual domination number \( \gamma(G) \).
A subset \( S \subseteq V(G) \) is said to be a total dominating set if every vertex in \( V(G) \) has at least one neighbor in \( S \), and it is a connected dominating set if the graph induced by \( S \) is connected. The total domination number \( \gamma_t(G) \) represents the cardinality of a minimum total dominating set of \( G \) and the connected domination number \( \gamma_c(G) \) the cardinality of a minimum connected dominating set.
Fink and Jacobson showed in 1985 that if \( G \) is a graph with \( \Delta(G) \geq p \geq 2 \), then \(\gamma_p(G) \geq \gamma(G) + p – 2.\)
In this paper, we will give some sufficient conditions for a graph \( G \) such that \(\gamma_p(G) \geq \gamma(G) + p – 1.\)
We will show that for block graphs \( G \) the inequality \(\gamma_p(G) \geq \gamma_t(G) + p – 2 \) is valid and that for trees \( T \) the inequality \(\gamma_p(T) \geq \gamma_c(T) + p – 1\) holds. Further, we characterize the trees \( T \) with \(\gamma_p(T) = \gamma_c(T) + p – 1,\) \(\gamma_p(T) = \gamma_t(T) + p – 2, \gamma_p(T) = \gamma_t(T) + p – 1,\) and \(\gamma_p(T) = \gamma(T) + p – 1.\)
Let \( G = (V, E) \) be a connected graph. A dominating set \( S \) of \( G \) is called a neighborhood connected dominating set (\({ncd-set}\)) if the induced subgraph \( \langle N(S) \rangle \) is connected. The minimum cardinality of an ncd-set of \( G \) is called the neighborhood connected domination number of \( G \) and is denoted by \( \gamma_{nc}(G) \). In this paper, we initiate a study of this parameter.
Let \( \lambda K_v \) be the complete multigraph, and \( G \) a finite simple graph. A \( G \)-design (\( G \)-packing, \( G \)-covering, respectively) of \( \lambda K_v \) is denoted by \( GD(v, G, \lambda) \) (\( PD(v, G, \lambda) \), \( CD(v, G, \lambda) \), respectively). In this paper, we will give some construction methods of graph packings and graph coverings, determine the existence spectrum for the \( G \)-designs of \( \lambda K_v \), and construct the maximum packings and the minimum coverings of \( \lambda K_v \) with \( G \) for any positive integer \( \lambda \). The graph \( G \) is either \( (K_4 – e) \cup P_1 \) or \( C_5 \bigodot P_1 \). Therefore, the problems of \( G \)-coverings and \( G \)-packings of \( \lambda K_v \) are solved completely when \( G \) is a graph with order \( 6 \) and \( |E(G)| \leq 6 \).
A maximal directed path in an acyclic orientation of a graph \( G \) is a path \( a_1 \to a_2 \to \cdots \to a_k \) such that \( \text{id}\; a_1 = \text{od}\; a_k = 0 \). The compression of \( G \) is the smallest integer \( k \) such that, for any acyclic orientation of \( G \), there is a maximal directed path of length at most \( k \). We characterize graphs with compression \( 1 \) and \( 2 \) and determine the compression of trees.
For each of the parameter sets \((30, 7, 15)\) and \((26, 12, 55)\), a simple \(3\)-design is given. They have \(\text{PSL}(2, 29)\) and \(\text{PSL}(2, 25)\) as their automorphism group, respectively. Each of the two simple \(3\)-designs is the first one ever known with the parameter set given and \(4\) in each of the two parameter sets is minimal for the given \(v\) and \(k\).
In this paper, we study linear codes over finite chain rings. We relate linear cyclic codes, \((1 + \gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over a finite chain ring \(R\), where \(\gamma\) is a fixed generator of the unique maximal ideal of the finite chain ring \(R\), and the nilpotency index of \(\gamma\) is \(k+1\). We also characterize the structure of \((1+\gamma^k)\)-cyclic codes and \((1 – \gamma^k)\)-cyclic codes over finite chain rings.
Let \(G\) be a graph with \(n\) vertices. The mean integrity of \(G\) is defined as follows:\(J(G) = min_{P \subseteq V} \{|P| + \tilde{m}(G – P)\},\) where \(\tilde{m}(G – P) = \frac{1}{n-|P|}\sum_{v \in G – P} n_v\) and \(n_v\) is the size of the component containing \(v\). The main result of this article is a formula for the mean integrity of a path \(P_n\) of \(n\) vertices. A corollary of this formula establishes the mean integrity of a cycle \(C_n\) of \(n\) vertices.
It is known that any reducible additive hereditary graph property has infinitely many minimal forbidden graphs, however the proof of this fact is not constructive. The purpose of this paper is to construct infinite families of minimal forbidden graphs for some classes of reducible properties. The well-known Hajós’ construction is generalized and some of its applications are presented.
In this paper, we prove that for any graph \(G\), there is a dominating induced subgraph which is a cograph. Two new domination parameters \(\gamma_{cd}\) – the cographic domination number and \(\gamma_{gcd}\) – the global cographic domination number are defined. Some properties, including complexity aspects, are discussed.
Given a permutation \(\pi\) chosen uniformly from \(S_n\), we explore the joint distribution of \(\pi(1)\) and the number of descents in \(\pi\). We obtain a formula for the number of permutations with \(Des(\pi) = d\) and \(\pi(1) = k\), and use it to show that if \(Des(\pi)\) is fixed at \(d\), then the expected value of \(\pi(1)\) is \(d+1\). We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when \(d\) is small the distribution of \(\pi(1)\) among the permutations with \(d\) descents is approximately geometric. Applications to Stein’s method and the Neggers-Stanley problem are presented.
A graph is called induced matching extendable, if every induced matching of it is contained in a perfect matching of it. A graph \(G\) is called \(2k\)-vertex deletable induced matching extendable, if \(G — S\) is induced matching extendable for every \(S \subset V(G)\) with \(|S| = 2k\). The following results are proved in this paper. (1) If \(\kappa(G) \geq \lceil \frac{v(G)}{3} \rceil +1\) and \(\max\{d(u), d(v)\} \geq \frac{2v(G)+1}{3}\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is induced matching extendable. (2) If \(\kappa(G) \geq \lceil \frac{v(G)+4k}{3}\rceil\) and \(\max\{d(u), d(v)\} \geq \lceil \frac{2v(G)+2k}{3} \rceil\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is \(2k\)-vertex deletable induced matching extendable. (3) If \(d(u) + d(v) \geq 2\lceil\frac{2v(G)+2k}{3} \rceil – 1\) for every two nonadjacent vertices \(u\) and \(v\), then \(G\) is \(2k\)-vertex deletable IM-extendable. Examples are given to show the tightness of all the conditions.
Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper, we determine the first three graphs among all bicyclic graphs with \(n\) vertices, ordered according to their least eigenvalues in increasing order.
The modified Zagreb indices are topological indices which reflect certain structural features of organic molecules. In this paper we study the modified Zagreb indices of joins and compositions.
In \([1]\), well-ordered Steiner triple systems were introduced and used to construct \(1\)-perfect partitions of the \(n\)-cube. However, non-trivial well-ordered Steiner triple systems were only known to exist when \(v =15\). In this short note, we present a simple construction to give a non-trivial well-ordered Steiner triple system of order \(v = 2^n – 1\) for all \(n \geq 5\) and this settles a problem in \([1]\).
Different neighbor conditions are considered in \([3,4,9]\) for a graph up-embeddable. In this paper, we consider the neighbor conditions of all the pairs of vertices with diameter \(2\) and obtain the following new result: if \(|N_G(u) \cap N_G(v)| \geq 2\) for any two vertices \(u,v \in D\) where \(D = \{(u, v) | d_G(u, v) = 2, u,v \in V(G)\}\), then \(G\) is up-embeddable.
We study the factorisations of a cyclic permutation of length \(n\) as a product of a minimal number of transpositions, calculating the number \(f(n, m)\) of factorisations in which a fixed element is moved \(m\) times. In this way, we also give a new proof-in the spirit of Clarke’s proof of Cayley’s theorem on the number of labelled trees-of the fact that there are \(n^{n-2}\) such factorisations.
We show that there are relationships between a generalized Lucas sequence and the permanent and determinant of some Hessenberg matrices.
Suppose \(G\) is a simple graph with average vertex degree greater than \(k – 2\). Erdős and Sós conjectured that \(G\) contains every tree on \(k\) vertices. Sidorenko proved \(G\) contains every tree that has a vertex \(v\) with at least \(\left\lfloor\frac{k}{2}\right\rfloor – 1\) leaf neighbors. We prove this is true if \(v\) has only \(\left\lceil\frac{k}{2}\right\rceil – 2\) leaf neighbors. We generalize Sidorenko’s result by proving that if \(G\) has minimum degree \(d\), then \(G\) contains every tree that has a vertex with at least \((k – 1) – d\) leaf neighbors. We use these results to prove that if \(G\) has average degree greater than \(k – 2\) and minimum degree at least \(k – 4\), then \(G\) contains every tree on \(k\) vertices.
A simple graph \(G\)is induced matching extendable, shortly IM-extendable, if every induced matching of \(G\) is included in a perfect matching of \(G\). The cyclic graph \(C_{2n}(1,k)\) is the graph with \(2n\) vertices \(x_0, x_1, \ldots, x_{2n-1}\), such that \(x_ix_j\) is an edge of \(C_{2n}(1,k)\) if either \(i-j \equiv \pm 1 \pmod{2n}\) or \(i-j \equiv \pm k \pmod{2n}\). We show in this paper that the only IM-extendable graphs in \(C_{2n}(1,k)\) are \(C_{2n}(1,3)\) for \(n \geq 4\); \(C_{2n}(1,n-1)\) for \(n \geq 3\); \(C_{2n}(1,n)\) for \(n \geq 2\); \(C_{2n}(1,\frac{n}{2})\) for \(n \geq 4\); \(C_{2n}(1,\frac{2n+1}{3})\) for \(n \geq 5\); \(C_{2n}(1,\frac{2n+2}{3})\) for \(n \leq 14\); \(C_{2n}(1,\frac{2n-2}{3})\) for \(n \leq 16\); \(C_{2n}(1,2)\) for \(n \leq 4\); \(C_{20}(1,8)\); \(C_{30}(1,6)\); \(C_{40}(1,8)\); \(C_{60}(1,12)\) and \(C_{80}(1,10)\).
For a vertex \(v\) in a graph \(G\), a local cut at \(v\) is a set of size \(d(v)\) consisting of the vertex \(x\) or the edge \(vx\) for each \(x \in N(v)\). A set \(U \subseteq V(G) \cup E(G)\) is a diameter-increasing set of \(G\) if the diameter of \(G – U\) is greater than the diameter of \(G\). In the present work, we first prove that every smallest generalized cutset of Johnson graph \(J(n,k)\) is a local cut except for \(J(4,2)\). Then we show that every smallest diameter-increasing set in \(J(n,k)\) is a subset of a local cut except for \(J(n,2)\) and \(J(6, 3)\).
Let \(G\) be a finite abelian group with exponent \(n\). Let \(s(G)\) denote the smallest integer \(l\) such that every sequence over \(G\) of length at least \(l\) has a zero-sum subsequence of length \(n\). For \(p\)-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on \(s(G)\), which allows to determine \(s(G)\) precisely in special cases. Our results contain Kemnitz’ conjecture, which was recently proved, as a special case.
Let \(\mathcal{D}\) be a \(2\)-\((v,k,4)\) symmetric design, and \(G\) be a subgroup of the full automorphism group of \(\mathcal{D}\). In this paper, we prove that if \(G \leq {Aut}(\mathcal{D})\) is flag-transitive, point-primitive then \(G\) is of affine or almost simple type. We prove further that if a nontrivial \(2\)-\((v, k, 4)\) symmetric design has a flag-transitive, point-primitive, almost simple automorphism group \(G\), then \(\text{Soc}(G)\) is not a sporadic simple group.
We prove explicit formulas for the rank polynomial and Whitney numbers of the distributive lattice of order ideals of the garland poset, ordered by inclusion.
A semi-double graph is such a connected multi-graph that each multi-edge consists of two edges. If there is at most one loop at each vertex of a semi-double graph, then this graph is called a single-petal graph. In this paper, we obtained that if \(G\) is a connected (resp. \(2\)-edge-connected, \(3\)-edge-connected) simple graph of order \(n\), then \(G\) is upper embeddable if \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-3}{2}\right\rceil\) (resp. \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-2}{3}\right\rceil, d_G(u) + d_G(v) \geq \left\lceil\frac{2n-23}{2}\right\rceil\)) for any two adjacent vertices \(u\) and \(v\) of \(G\). In addition, by means of semi-double graph and single-petal graph, the upper embeddability of multi-graph and pseudograph are also discussed in this paper.
Let \(d(n, k)\) denote the number of derangements (permutations without fixed points) with \(k\) cycles of the set \([n] = \{1, 2, \ldots, n\}\). In this paper, a new explicit expression for \(d(n, k)\) is presented by graph theoretic method, and a concise regular binary tree representation for \(d(n, k)\) is provided.
This paper devotes to the investigation of \(3\)-designs admitting the special projective linear group \(\text{PSL}(2,q)\) as an automorphism group. When \(q \equiv 3 \pmod{4}\), we determine all the possible values of \(\lambda\) in the simple \(3\)-\((q+1, 7, \lambda)\) designs admitting \(\text{PSL}(2,q)\) as an automorphism group.
We give an optimal degree condition for a tripartite graph to have a spanning subgraph consisting of complete graphs of order \(3\). This result is used to give an upper bound of \(2\Delta\) for the strong chromatic number of \(n\) vertex graphs with \(\Delta \geq n/6\).
A partial Latin square \(P\) of order \(n\) is an \(n \times n\) array with entries from the set \(\{1, 2, \ldots, n\}\) such that each symbol is used at most once in each row and at most once in each column. If every cell of the array is filled, we call \(P\) a Latin square. A partial Latin square \(P\) of order \(n\) is said to be avoidable if there exists a Latin square \(L\) of order \(n\) such that \(P\) and \(L\) are disjoint. That is, corresponding cells of \(P\) and \(L\) contain different entries. In this note, we show that, with the trivial exception of the Latin square of order \(1\), every partial Latin square of order congruent to \(1\) modulo \(4\) is avoidable.
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 \(\{a_ib_j : 0 \leq i \leq n-1, j = i+1, i+2, \ldots, i+k \pmod{n}\}\). A caterpillar is a tree of order at least three which contains a path such that each vertex not on the path is adjacent to a vertex on the path. Being a connected bipartite graph, a caterpillar is balanced if the two parts of the bipartition of its vertices have equal size; otherwise, it is unbalanced. In this paper, we obtain the necessary and sufficient condition for balanced-caterpillar factorization of crowns. The criterion for unbalanced-caterpillar factorization of crowns is open. We also obtain the necessary and sufficient condition for directed caterpillar factorization of symmetric crowns.
This paper determines that the connectivity of the Cartesian product \(G_1 \square G_2\) of two graphs \(G_1\) and \(G_2\) is equal to \(\min\{\kappa_1v_2 + \kappa_2v_1, \delta_1 + \delta_2 \}\), where \(v_i, \kappa_i\), and \(\delta_i\) are the order, connectivity, and minimum degree of \(G_i\), respectively, for \(i = 1, 2\). Additionally, some necessary and sufficient conditions are given for \(G_1 \square G_2\) to be maximally connected and super-connected.
This paper deals with two types of graph labelings namely, super \((a, d)\)-edge antimagic total labeling and \((a, d)\)-vertex antimagic total labeling. We provide super \((a, d)\)-edge antimagic total labeling for disjoint unions of Harary graphs and disjoint unions of cycles. We also provide \((a,d)\)-vertex antimagic total labeling for disjoint unions of Harary graphs, disjoint unions of cycles, sun graphs and disjoint unions of sun graphs,
The existence question for a \(3\)-\((16,7,5)\) design is open, In this paper, we examine possible automorphisms of this design. We consider a minimum subset of basic permutations consisting of cycles of prime length \(p\) and prove that if a \(3\)-\((16,7,5)\) design exists, then it is either rigid or admits basic automorphisms with cycles of length \(2\) or \(3\).
We define a product summation of ordered partition \(f_j(n,m,r) = \sum{c_1^r c_2^r \ldots c_j^rc_{j+1} \ldots c_m}\), where the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\) and \(0 \leq j \leq m\). We concentrate on \(f_m(n,m,r)\) in this paper. The main results are as follows:
(1) The generating function for \(f_m(n,m,r)\) and the explicit formula for \(f_m(n,m,2) , f_m(n,m,3)\) and \(f_m(n,m, 4)\) are obtained.
(2) The relationship between \(f_j(n,m,r)\) for \(r = 2,3\) and the Fibonacci and Lucas numbers is found.
It is shown that for \(2 \leq t \leq n-3\), a strict \(t\)-SB\((n,n-1)\) design does not exist, but for \(n \geq 3\), a non-strict \(2\)-SB\((n,n-1)\) design exists. The concept of large sets for Steiner triple systems is extended to SB designs and examples of large sets for SB designs are given.
It is shown that every well-defined solution to the second-order difference equation in the title, when \((A_n)_{n \in 0}\) is a two-periodic sequence such that \(\max\{A_0, A_1\} \geq 0\), is eventually periodic with period two. In the case \(\max\{A_0, A_1\} \leq 0\), it is shown the existence of unbounded solutions, by describing all solutions in terms of \(A_0\), \(A_1\), \(x_{-1}\), and \(x_0\).
This paper considers the folded hypercube \(FQ_n\) as an enhancement on the hypercube, and obtains some algebraic properties of \(FQ_n\). Using these properties, the authors show that for any two vertices \(x\) and \(y\) in \(FQ_n\), with distance \(d\) and any integers \(h \in \{d, n+1- d\}\) and \(l\) with \(h \leq l \leq 2^n – 1\), \(FQ_n\) contains an \(xy\)-path of length \(l\) and no \(xy\)-path of other length, provided that \(l\) and \(h\) have the same parity.
Let \(G\) be a \(2\)-tough graph on at least five vertices and let \(e_1, e_2\) be a pair of arbitrarily given edges of \(G\). Then
(a) There exists a \(2\)-factor in G containing \(e_1, e_2\).
(b) There exists a \(2\)-factor in G avoiding \(e_1, e_2\).
(c) There exists a \(2\)-factor in G containing \(e_1\) and avoiding \(e_2\).
In this paper, a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations
\[x_{n+1} = \frac{x_ny_{n-1}}{x_ny_{n-1}+1} ,y_{n+1}=\frac{y_n x_{n-1}}{y_nx_{n-1} + 1} , \quad n = 0, 1, 2, \ldots,\]
where the initial values \((x_k, y_k) \in (0, \infty) (\text{for} k=-1,0)\).
Erdős and Sós conjectured in \(1962\) that if the average degree of a graph \(G\) exceeds \(k – 2\), then \(G\) contains every tree on \(k\) vertices. Results from Sauer and Spencer (and independent results from Zhou) prove the special case where \(G\) has \(k\) vertices. Results from Slater, Teo, and Yap prove the case where \(G\) has \(k + 1\) vertices. In \(1996\), Woźniak proved the case where \(G\) has \(k + 2\) vertices. We prove the conjecture for the case where \(G\) has \(k + 3\) vertices.
A semi-double graph is a connected multi-graph such that each multi-edge consists of two edges. If there is at most one loop at each vertex of a semi-double graph, then this graph is called a single-petal graph. Via the degree-sum of nonadjacent vertices, the up-embeddability of semi-double graphs and single-petal graphs are discussed in this paper. And the results obtained in this paper can be extended to determine the up-embeddability of multi-graphs and pseudographs.
The commuting graph of an arbitrary ring \(R\), denoted by \(\Gamma(R)\), is the graph whose vertices are all non-central elements of \(R\), and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = ba\). In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of the quaternion algebra \(\mathbb{Z}_n[i, j, k]\).
A set \(D\) of vertices of a graph \(G = (V, E)\) is a \(\textit{dominating set}\) if every vertex of \(V-D\) is adjacent to at least one vertex in \(D\). The \(\textit{domination number}\) \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A subset of \(V-D\), which is also a dominating set of \(G\), is called an \(\textit{averse dominating set}\) of \(G\) with respect to \(D\). The \(\textit{inverse domination number}\) \(\gamma'(G)\) equals the minimum cardinality of an inverse dominating set \(D\). In this paper, we study classes of graphs whose domination and inverse domination numbers are equal.
A strongly connected digraph \(\Gamma\) is said to be walk regular if for any nonnegative integer \(l\) and any vertex \(u\) of \(\Gamma\), the number of circuits of length \(l\) containing \(u\) depends only on \(l\). This family of digraphs is a directed version of walk regular graphs. In this paper, we discuss some basic properties of walk regular digraphs.
For any \(n \geq 2\) we let \(S_n\) be the set of permutations of the set \(\{1,2,\ldots,n\}\). A reduction \(\overline{f}\) on \(S_n\) is a set of functions \(\{f_i : 1 \leq i \leq n\}\) such that \(f_n\) is the identity function on \(\{1,2,\ldots,n-1\}\) and for \(i n_0\), such that \(\phi(n) \leq n\) for all \(n \geq n_0\), and for which \(p \downarrow \phi(n) \downarrow i = p \downarrow i \downarrow n-1\) for all \(n > n_0\), for all \(i \leq n-1\) and for all \(p \in S_n\). And the system is said to be amenable if for every \(n > n_0\) there is an integer \(k < n\) such that, for all \(p \in S_n\), \(p \downarrow k \downarrow n-1 = p \downarrow n-1\). The purpose of this paper is to study faithful reductions and linked reduction systems. We characterize amenable, linked reduction systems by means of two types of liftings by which a reduction on \(S_{n+1}\) can be formed from one on \(S_n\). And we obtain conditions for a reduction system to be faithful. One interesting consequence is that any amenable, linked reduction system which begins with a simple reduction is faithful.
Let \(N\) be a positive integer and let \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_l)\) be a partition of \(N\) of length \(l\), i.e., \(\sum_{i=1}^{l}\lambda_i = N\) with parts \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_l \geq 1\). Define \(T(\lambda)\) as the partition of \(N\) with parts \(l\), \(\lambda_1 – 1, \lambda_2 – 1, \ldots, \lambda_l – 1\), ignoring any zeros that might occur. Starting with a partition \(\lambda\) of \(N\), we describe Bulgarian Solitaire by repeatedly applying the shift operation \(T\) to obtain the sequence of partitions
\[\lambda, T(\lambda), T^2(\lambda), \ldots\]
We say a partition \(\mathcal{A}\) of \(N\) is \(T\)-cyclic if \(T^i(\mu) = \mu\) for some \(i \geq 1\). Brandt \([2]\) characterized all \(T\)-cyclic partitions for Bulgarian Solitaire. In this paper, we give an inductive proof of Brandt’s result.
Let \(D\) be an acyclic digraph. The competition graph of \(D\) is a graph which has the same vertex set as \(D\) and has an edge between \(x\) and \(y\) if and only if there exists a vertex \(v\) in \(D\) such that \((x, v)\) and \((y, v)\) are arcs of \(D\). For any graph \(G\), \(G\) together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number \(k(G)\) of \(G\) is the smallest number of such isolated vertices.
A hole of a graph is a cycle of length at least \(4\) as an induced subgraph. In \(2005\), Kim \([5]\) conjectured that the competition number of a graph with \(h\) holes is at most \(h + 1\). Though Li and Chang \([8]\) and Kim et al. \([7]\) showed that her conjecture is true when the holes do not overlap much, it still remains open for the case where the holes share edges in an arbitrary way. In order to share an edge, a graph must have at least two holes and so it is natural to start with a graph with exactly two holes. In this paper, the conjecture is proved true for such a graph.
Let \(G\) be a graph with domination number \(\gamma(G)\). A dominating set \(S \subseteq V(G)\) has property \(\mathcal{UK}\) if all components of the subgraph it induces in \(G\) are complete. The union of complete graphs domination number of a graph \(G\), denoted \(\gamma_{uk}(G)\), is the minimum possible size of a dominating set of \(G\), which has property \(\mathcal{UK}\). Results on changing and unchanging of \(\gamma_{uk}(G)\) after vertex removal are presented. Also forbidden subgraph conditions sufficient to imply \(\gamma(G) = \gamma_{uk}(G)\) are given.
In this paper, we use the finite Heine \({}_{2}\Phi_1\) transformations given in \([4]\) and some elementary simplifications to obtain several Rogers-Ramanujan type identities.
Using lines in a two-dimensional vector space \({GF}(q^2)\) over \({GF}(q)\), we construct some classes of external difference families over \({GF}(q^2)\).
A digraph \(D\) is a local out-tournament if the outset of every vertex is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, \(n\). Corresponding results for biclique covers and partitions of the digraph are provided.
For an ordered set \( W = \{w_1, w_2, \dots, w_k\} \) of \( k \) distinct vertices in a nontrivial connected graph \( G \), the metric code of a vertex \( v \) of \( G \) with respect to \( W \) is the \( k \)-vector
\[
\text{code}(v) = (d(v, w_1), d(v, w_2), \dots, d(v, w_k)),
\]
where \( d(v, w_i) \) is the distance between \( v \) and \( w_i \) for \( 1 \leq i \leq k \). The set \( W \) is a local metric set of \( G \) if \( \text{code}(u) \neq \text{code}(v) \) for every pair \( u, v \) of adjacent vertices of \( G \). The minimum positive integer \( k \) for which \( G \) has a local metric set of cardinality \( k \) is the local metric dimension \(\text{lmd}(G)\) of \( G \). We determine the local metric dimensions of joins and compositions of some well-known classes of graphs, namely complete graphs, cycles, and paths. For a nontrivial connected graph \( G \), a vertex \( v \) of \( G \), and an edge \( e \) of \( G \), where \( v \) is not a cut-vertex and \( e \) is not a bridge, it is shown that \(\text{lmd}(G – v) \leq \text{lmd}(G) + \text{deg}(v)\) and \(\text{lmd}(G – e) \leq \text{lmd}(G) + 2.
\) The sharpness of these two bounds is studied. We also present several open questions in this area of research.
Let \( G \) be a connected graph. In this paper, we introduce the concepts of vertex-to-clique \({radius}\) \( r_1 \), vertex-to-clique \({diameter}\) \( d_1 \), clique-to-vertex \({radius}\) \( r_2 \), clique-to-vertex \({diameter}\) \( d_2 \), clique-to-clique \({radius}\) \( r_3 \), and clique-to-clique \({diameter}\) \( d_3 \) in \( G \). We prove that for any connected graph, \( r_i \leq d_i \leq 2r_i + 1 \) for \( i = 1, 2, 3 \). We also find expressions for \( d_1 \), \( d_2 \), and \( d_3 \) for a tree \( T \) in terms of \( r_1 \), \( r_2 \), and \( r_3 \) respectively, which determine the cardinality of each \( Z_i(T) \), where \( Z_i(T) \) is the vertex-to-clique, the clique-to-vertex, and the clique-to-clique center respectively of \( T \) for \( i = 1, 2, 3 \). If \( G \) is a graph that is not a tree and if \( g(G) \) denotes the girth of the graph, then its relation with each of \( d_1 \), \( d_2 \), and \( d_3 \) is discussed. We also characterize the class of graphs \( G \) such that \( G \) is not a tree, \( d_3 \neq 0 \), and \( g(G) = 2d_3 + 3 \).
A tournament is an orientation of a complete graph, and a multipartite or \( c \)-partite tournament is an orientation of a complete \( c \)-partite graph. If we speak of a path, then we mean a directed path.
Let \( D \) be a regular \( c \)-partite tournament with \( r \) vertices in each partite set, and let \( X \subseteq V(D) \) be an arbitrary set with exactly \( 2 \) vertices from each partite set. For all \( c \geq 4 \), the authors determined in a recent article the minimal value \( g(c) \) such that \( D – X \) is Hamiltonian for every regular multipartite tournament with \( r \geq g(c) \). In this paper, we will supplement this result by postulating a given path covering number instead of the Hamiltonicity of the digraph \( D – X \). This means, for all \( c \geq 4 \) and \( k \geq 1 \), we will determine the minimal value \( h(k, c) \) such that \( D – X \) can be covered by at most \( k \) paths for every regular \( c \)-partite tournament with \( r \geq h(k, c) \). Moreover, we will present the minimal path covering number of \( D – X \), if \( D \) is a regular \( 3 \)-partite tournament and \( X \) contains exactly \( s \) vertices (\( s \geq 2 \)) from every partite set.
We investigate the problem of decomposing the edges of a connected circulant graph with \( n \) vertices and generating set \( S \) into isomorphic subgraphs, each having \( n \) edges. For \( 8 \)-regular circulants, we show that this is always possible when \( s+2 \leq \frac{n}{4} \) for all edge lengths \( s \in S \).
This paper continues the results of “Domination Cover Pebbling: Graph Families.” An almost sharp bound for the domination cover pebbling (DCP) number, \( \psi(G) \), for graphs \( G \) with specified diameter has been computed. For graphs of diameter two, a bound for the ratio between \( \lambda(G) \), the cover pebbling number of \( G \), and \( \psi(G) \) has been computed. A variant of domination cover pebbling, called subversion DCP, is introduced, and preliminary results are discussed.
A graph \(G\) has a representation modulo \(n\) if there exists an injective map \(f: V(G) \to \{0, 1, \dots, n-1\}\) such that vertices \(u\) and \(v\) are adjacent if and only if \(f(u) – f(v)\) is relatively prime to \(n\). The representation number \(rep(G)\) is the smallest \(n\) such that \(G\) has a representation modulo \(n\). In 2000, Evans, Isaak, and Narayan determined the representation number of a complete graph minus a path. In this paper, we refine their methods and apply them to the family of complete graphs minus a disjoint union of paths.
Let \(G = (V, E)\) be a graph and \(\overline{G}\) be the complement of \(G\). The complementary prism of \(G\), denoted \(G \overline{G}\), is the graph formed from the disjoint union of \(G\) and \(\overline{G}\) by adding the edges of a perfect matching between the corresponding vertices of \(G\) and \(\overline{G}\). A set \(D \subseteq V(G)\) is a locating-dominating set of \(G\) if for every \(u \in V(G) \setminus D\), its neighborhood \(N(u) \cap D\) is nonempty and distinct from \(N(v) \cap D\) for all \(v \in V(G) \setminus D\) where \(v \neq u\). The locating-domination number of \(G\) is the minimum cardinality of a locating-dominating set of \(G\). In this paper, we study locating-domination of complementary prisms. We determine the locating-domination number of \(G \overline{G}\) for specific graphs \(G\) and characterize the complementary prisms with small locating-domination numbers. We also present upper and lower bounds on the locating-domination numbers of complementary prisms, and we show that all values between these bounds are achievable.
A disjoint multiple paths problem asks if there exist paths between a given set of vertices. Constraints are applied so that paths are not allowed to share vertices (vertex disjoint multiple paths) or share edges (edge disjoint multiple paths). The vertex disjoint multiple paths problem is one of the classic NP-complete problems presented by Karp [1]. The edge disjoint multiple paths problem is also NP-complete since it is easily transformed from the vertex disjoint multiple paths problem. Because of its importance in electronic circuit design, studies are done for restricted cases. The edge disjoint multiple paths problem remains NP-complete for acyclic graphs and planar graphs. Furthermore, the edge disjoint multiple paths problem remains NP-complete if the graph is limited to an undirected mesh.
In this paper, the edge disjoint multiple paths problem when constructed over a directed mesh is discussed. We found that the multiple paths problem remains NP-complete in this special case. Three polynomial time algorithms are presented in which the following restrictions are made: (i) disjoint paths with the same origin row, the same destination row, distinct origin columns, and distinct destination columns, (ii) disjoint paths with the same origin column, the same destination column, distinct origin rows, and distinct destination rows, and (iii) disjoint paths with the same origin row, distinct origin columns, and distinct destination rows.
We discuss a transform on the set of rational functions over the finite field \( \mathbb{F}_q \). For a subclass of these functions, the transform yields a polynomial and its factorization as a product of the set of monic irreducible polynomials, all of which share a common property \( P \) that depends on the choice of rational function. A general formula is derived from the factorization for the number of monic irreducible polynomials of degree \( n \) having property \( P \). However, it is also possible in some instances to exploit the properties of the factorization to obtain a “closed” form of the answer more directly. We illustrate the method with four examples, two of which appear in the literature. In particular, we give alternative proofs for a result of L. Carlitz on the number of monic irreducible self-reciprocal polynomials and a remarkable result of S. D. Cohen on the number of \((r, m)\)-polynomials, that is, monic irreducible polynomials of the form \( f(x^r) \) of degree \( mr \). We also give a generalization of the factorization of \( x^{q-1} – 1 \) over \( \mathbb{F}_q \) that includes the factorization of \( x^{(q-1)^2} – 1 \). The new results concern translation invariant polynomials, which lead to a consideration of the orders of elements in \( \overline{\mathbb{F}}_q \), the algebraic closure of \( \mathbb{F}_q \). We show that there are an infinite number of \( \theta \in \overline{\mathbb{F}}_q \) such that \( \text{ord}(\theta) \) and \( \text{ord}(r(\theta)) \) are related, in the sense that given one, one can infer information about the other.
Let \( G \) be a graph with \( v \) vertices. If there exists a collection of lists of colors \(\{S_1, S_2, \ldots, S_v\}\) on its vertices, each of size \( k \), such that there exists a unique proper coloring for \( G \) from this list of colors, then \( G \) is called a \({uniquely \;k \;-list\; colorable \;graph}\). In this note, we present a uniquely \( 3 \)-list colorable, planar, and \( K_4 \)-free graph. It is a counterexample to a conjecture by Ch. Eslahchi, M. Ghebleh, and H. Hajiabolhassan [3].
For any integers \( k, d \geq 1 \), a \((p, q)\)-graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), where \( p = |V(G)| \) and \( q = |E(G)| \), is said to be \((k, d)\)-strongly indexable (in short \((\textbf{k, d})\)-\textbf{SI}) if there exists a pair of functions \((f, f^+)\) that assigns integer labels to the vertices and edges, i.e., \( f: V(G) \to \{0, 1, \dots, p-1\} \) and \( f^+: E(G) \to \{k, k+d, k+2d, \dots, k+(q-1)d\} \), such that \( f^+(u, v) = f(u) + f(v) \) for any \((u, v) \in E(G)\). We determine here classes of spiders that are \((1, 2)\)-SI graphs. We show that every given \((1, 2)\)-SI spider can be extended to an \((1, 2)\)-SI spider with arbitrarily many legs.
In this paper, we obtain some new results, using inequalities such as Hölder and Minkowski, etc., on the existence of balanced arrays (B-arrays) with two levels and of strength six. We then discuss the use of these results to obtain the maximum number of constraints for B-arrays with given values of the parameter vector \(\underline{\mu}’\). We also include some illustrative examples.
A construction of a minimum cycle basis for the wreath product of a star by a path, two stars and a star by a wheel is given. Moreover, the basis numbers of these products are determined.
For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h \setminus \{0\} \) such that the induced vertex labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by
\[ l^+(v) = \sum_{uv \in E(G)} l(uv), \]
is a constant map. When this constant is \( 0 \), we call \( G \) a zero-sum \( h \)-magic graph. The null set of \( G \) is the set of all natural numbers \( h \in \mathbb{N} \) for which \( G \) admits a zero-sum \( h \)-magic labeling. A graph \( G \) is said to be uniformly null if every magic labeling of \( G \) induces a zero sum. In this paper, we will identify the null sets of certain planar graphs such as wheels and fans.
In this paper, we find six new weighing matrices of order \( 2n \) and weight \( 2n-9 \) constructed from two circulants, by establishing various patterns on the locations of the nine zeros in a potential solution.
In the past few years, several studies have appeared that relate to the existence of \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments. In these studies, the number of players in the tournament is taken to be a prime \( p \) of the form \( p \equiv 2^k + 1 \pmod{2^k+1} \), where \( k \geq 2 \). For the cases \( k = 2, 3, 4 \) it has been shown [6,4,5,12] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes except for the impossible cases \( p = 5, 13, 17 \). For the cases \( k = 5, 6, 7 \) it has been shown [13] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) and that \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) with the exception that existence or non-existence of these designs for \( p = 97, 193, 449, 577, 641, 1409 \) is an open question. Here the case \( k = 8 \) is considered. It is established that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all primes \( p \equiv 257 \pmod{512} \), \( p \leq 6{,}944{,}177 \), except possibly for \( p = 257, 769, 3329 \). For \( p = 3329 \) we are able to construct a \( \mathbb{Z} \)-cyclic directed-triplewhist tournament, but the existence of a \( \mathbb{Z} \)-cyclic ordered-triplewhist tournament remains an open question. Furthermore, for each type of design it is conjectured that our basic constructions will produce these designs whenever \( p > 5{,}299{,}457 \).
The domination graph of a digraph \( D \), denoted \( \text{dom}(D) \), is created using the vertex set of \( D \) and edge \( uv \in E(\text{dom}(D)) \) whenever \( (u,z) \in A(D) \) or \( (v,z) \in A(D) \) for any other vertex \( z \in V(D) \). Specifically, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. In particular, digraphs are completely characterized where \( UG^c(D) \) is the union of two disjoint paths.
The current paper deals with two special matrices \(T_n\) and \(W_n\) related to the Pascal, Vandermonde, and Stirling matrices. As a result, various properties of the entries of \(T_n\) and \(W_n\) are obtained, including the generating functions, recurrence relations, and explicit expressions. Some additional results are also presented.
There are some results and many conjectures with the conclusion that a graph \(G\) contains all trees of given size \(k\). We prove some new results of this type.
In \([3]\), we gave a factorization of the generalized Lah matrix.In this short note, we show its another factorization. From this factorization, several interesting combinatorial identities involving the Fibonacci numbers are obtained.
Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-decomposition of \(K_v\), denoted by \(G-GD_\lambda(v)\), is a pair \((X, \mathcal{B})\) where \(X\) is the vertex set of \(K_v\) and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, nine graphs \(G_i\) with six vertices and nine edges are discussed, and the existence of \(G_i-GD_\lambda(v)\) is given, \(1 \leq i \leq 9\).
Let \(G = (V, E)\) be a graph. A set \(S \subseteq V\) is a restrained dominating set if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V – S\). The restrained domination number of \(G\), denoted by \(\gamma_r(G)\), is the smallest cardinality of a restrained dominating set of \(G\). It is known that if \(T\) is a tree of order \(n\), then \(\gamma_r(T) \geq \left\lceil \frac{n+2}{3} \right\rceil\). In this note, we provide a simple constructive characterization of the extremal trees \(T\) of order \(n\) achieving this lower bound.
Given non-negative integers \(r, s\), and \(t\), an \([r, s, t]\)-coloring of a graph \(G = (V(G), E(G))\) is a mapping \(c\) from \(V(G) \cup E(G)\) to the color set \(\{0, 1, \ldots, k-1\}\) such that \(|c(v_i) – c(v_j)| \geq r\) for every two adjacent vertices \(v_i, v_j\), \(|c(e_i) – c(e_j)| \geq s\) for every two adjacent edges \(e_i, e_j\), and \(|c(v_i) – c(e_i)| \geq t\) for all pairs of incident vertices and edges, respectively. The \([r, s, t]\)-chromatic number \(\chi_{r,s,t}(G)\) of \(G\) is defined to be the minimum \(k\) such that \(G\) admits an \([r, s, t]\)-coloring. We prove that \(\chi_{1,1,2}(K_5) = 7\) and \(\chi_{1,1,2}(K_6) = 8\).
We determine a recursive formula for the number of rooted complete \(N\)-ary trees with \(n\) leaves, which generalizes the formula for the sequence of Wedderburn-Etherington numbers. The diagonal sequence of our new sequences equals the sequence of numbers of rooted trees with \(N + 1\) vertices.
In this paper, we determine the conics characterizing the generalized Fibonacci and Lucas sequences with indices in arithmetic progressions, generalizing work of Melham and McDaniel.
A graph \(G = (V, E)\) is a mod sum graph if there exists a positive integer \(z\) and a labeling, \(\lambda\), of the vertices of \(G\) with distinct elements from \(\{1, 2, \ldots, z-1\}\) such that \(uv \in E\) if and only if the sum, modulo \(z\), of the labels assigned to \(u\) and \(v\) is the label of a vertex of \(G\). The mod sum number \(\rho(G)\) of a connected graph \(G\) is the smallest nonnegative integer \(m\) such that \(G \cup mK_1\), the union of \(G\) and \(m\) isolated vertices, is a mod sum graph. In Section \(2\), we prove that \(F_n\) is not a mod sum graph and give the mod sum number of \(F_n\) (\(n \geq 6\) is even). In Section \(3\), we give the mod sum number of the symmetric complete graph.
In this paper, we consider the effect of edge contraction on the domination number and total domination number of a graph. We define the (total) domination contraction number of a graph as the minimum number of edges that must be contracted in order to decrease the (total) domination number. We show that both of these two numbers are at most three for any graph. In view of this result, we classify graphs by their (total) domination contraction numbers and characterize these classes of graphs.
In this paper, connected graphs with the largest Laplacian eigenvalue at most \(\frac{5+\sqrt{13}}{2}\) are characterized. Moreover, we prove that these graphs are determined by their Laplacian spectrum.
An extended directed triple system of order \(v\) with an idempotent element (EDTS(\(v, a\))) is a collection of triples of the type \([x, y, z]\), \([x, y, x]\) or \((x, x, x)\) chosen from a \(v\)-set, such that every ordered pair (not necessarily distinct) belongs to only one triple and there are \(a\) triples of the type \((x, x, x)\). If such a design with parameters \(v\) and \(a\) exists, then it will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). A necessary and sufficient condition for the existence of EDTS(\(v, 0\)) and EDTS(\(v, 1\)) are \(v \equiv 0 \pmod{3}\) and \(v \not\equiv 0 \pmod{3}\), respectively. In this paper, we have constructed two EDTS(\(v, a\))’s such that the number of common triples is in the set \(\{0, 1, 2, \ldots, b_{v,a} – 2, b_{v,a}\}\), for \(a = 0, 1\).
As applications of the Anzahl theorems in finite orthogonal spaces, we study the critical problem of totally isotropic subspaces, and obtain the critical exponent.
Let \(P(G,\lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H,\lambda) = P(G, \lambda)\) implies H is isomorphic to \(G\). In this paper, we study the chromaticity of Turén graphs with deleted edges that induce a matching or a star. As a by-product, we obtain new families of chromatically unique graphs.
Let \(\{w_n\}\) be a second-order recurrent sequence. Several identities about the sums of products of second-order recurrent sequences were obtained and the relationship between the second-order recurrent sequences and the recurrence coefficient revealed. Some identities about Lucas sequences, Lucas numbers, and Fibonacci numbers were also obtained.
In this paper, we prove that for any positive integers \(k,n\) with \(k \geq 2\) , the graph \(P_k^n\) is a divisor graph if and only if \(n \leq 2k + 2\) , where \(P^k_n\) is the \(k\) th power of the path \(P_n\). For powers of cycles we show that \(C^k_n\) is a divisor graph when \(n \leq 2k + 2\), but is not a divisor graph when \(n \geq 2k + 2\),but is not a divisor graph when \(n\geq 2k+\lfloor \frac{k}{2}\rceil,\) where \(C^k_n\) is the \(k\)th power of the cycle \(C_n\). Moreover, for odd \(n\) with \(2k+2 < n < 2k + \lfloor\frac{k}{2}\rfloor + 3\), we show that the graph \(C^k_n\) is not a divisor graph.
The Wiener index of a graph \(G\) is defined as \(W(G) = \sum_{u,v \in V(G)} d_G(u,v),\) where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\) and the sum goes over all pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the minimal and maximal Wiener index.
In this paper, we consider a random mapping, \(\hat{T}_n\), of the finite set \(\{1,2,\ldots,n\}\) into itself for which the digraph representation \(\hat{G}_n\) 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 a random permutation to the selected cyclic vertices.We investigate \(\hat{k}_n\), the size of a `typical’ component of \(\hat{G}_n\), and, under the assumption that the random permutation on the cyclical vertices is uniform, we obtain the asymptotic distribution of \(k\), 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 \(\hat{G}_n\) converges to the Poisson-Dirichlet \((1)\) distribution as \(n \to \infty\). Other applications and generalizations are also discussed in Section \(3\).
De Bruijn digraphs and shuffle-exchange graphs are useful models for interconnection networks. They can be represented as group action graphs of the wrapped butterfly graph and the cube-connected cycles, respectively. The Kautz digraph has similar definitions and properties to de Bruijn digraphs. It is \(d\)-regular and strongly \(d\)-connected, thus it is a group action graph. In this paper, we use another representation of the Kautz digraph and settle the open problem posed by M.-C. Heydemann in \([6]\).
We discuss the use of \(K\)-terminal networks to represent arbitrary clutters. A given clutter has many different representations, and there does not seem to be any set of simple transformations that can be used to transform one representation of a clutter into any other. We observe that for \(t \geq 2\) the class of clutters that can be represented using no more than \(t\) terminals is closed under minors, and has infinitely many forbidden minors.
Brenti (J. Combin. Theory Ser. A \(91 (2000))\) considered a \(q\)-analogue of the Eulerian polynomials by enumerating permutations in the symmetric group \(S_n\) with respect to the numbers of excedances and cycles. Here we establish a connection between these \(q\)-Eulerian polynomials and some infinite generating functions.
Let \(K_k, C_k, T_k\), and \(P_k\) denote a complete graph on \(k\) vertices, a cycle on \(k\) vertices, a tree on \(k+1\) vertices, and a path on \(k+1\) vertices, respectively. Let \(K_m-H\) be the graph obtained from \(K_m\) by removing the edges set \(E(H)\) of the graph \(H\) (\(H\) is a subgraph of \(K_m\)). A sequence \(S\) is potentially \(K_m-H\)-graphical if it has a realization containing a \(K_m-H\) as a subgraph. Let \(\sigma(K_m-H,n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma(S) \geq \sigma(K_m-H,n)\) is potentially \(K_m-H\)-graphical. In this paper, we determine the values of \(\sigma(K_{r+1}-H,n)\) for \(n \geq 4r+10, r \geq 3, r+1 \geq k \geq 4\) where \(H\) is a graph on \(k\) vertices which contains a tree on \(4\) vertices but not contains a cycle on \(3\) vertices. We also determine the values of \(\sigma(K_{r+1}-P_{2},n)\) for \(n \geq 4r+8, r \geq 3\).
In this paper, the class of \((m,n)\)-ary hypermodules is introduced and several properties and examples are found. \((m,n)\)-ary hypermodules are a generalization of hypermodules. On the other hand, we can consider \((m,n)\)-ary hypermodules as a good generalization of \((m,n)\)-ary modules. We define the fundamental relation \(\epsilon^*\) on the \((m,n)\)-ary hypermodules \(M\) as the smallest equivalence relation such that \(M/\epsilon^*\) is an \((m,n)\)-ary module, and then some related properties are investigated.
For given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1, G_2)\) is defined to be the least positive integer \(n\) such that every graph \(G\) on \(n\) vertices, either \(G\) contains a copy of \(G_1\) or the complement of \(G\) contains a copy of \(G_2\). In this note, we show that \(R(C_m, B_n) = 2m-1\) for \(m \geq 2n-1 \geq 7\). With the help of computers, we obtain the exact values of \(14\) small cycle-book Ramsey numbers.
For positive integers \(c \geq 0\) and \(k \geq 1\), let \(n = R(c, k)\) be the least integer, provided it exists, such that every \(2\)-coloring of the set \([1,n] = \{1,\ldots,n\}\) admits a monochromatic solution to the equation \(x + y+c = 4z\) with \(x, y, z \in [1,n]\). In this paper, the precise value of \(R(c, 4)\) is shown to be \(\left\lceil{3c + 2}/{8}\right\rceil\) for all even \(c \geq 34\).
Given a positive integer \(n\) such that \(-1\) is a quadratic residue mod \(n\), we give an algorithm that computes the integers \(u\) and \(v\) which satisfy the equation \(n = u^2 + v^2\). To do this, we use the group structure of the Modular group \(\Gamma= \text{PSL}(2,\mathbb{Z})\).
For a graph \(G = (V(G),E(G))\), a set \(S \subseteq V(G)\) is called a dominating set if \(N_G[S] = V(G)\). A dominating set \(S\) is said to be minimal if no proper subset \(S’ \subset S\) is a dominating set. Let \(\gamma(G)\) (called the domination number) and \(\Gamma(G)\) (called the upper domination number) be the minimum cardinality and the maximum cardinality of a minimal dominating set of \(G\), respectively. For a tree \(T\) of order \(n \geq 2\), it is obvious that \(1 = \gamma(K_{1,n-1}) \leq \gamma(T) \leq \Gamma(T) \leq \Gamma(K_{1,n-1}) = n-1\). Let \(t(n) = \min_{|T|=n}(\Gamma(T)-\gamma(T))\). In this paper, we determine \(t(n)\) for all natural numbers \(n\). We also characterize trees \(T\) with \(\Gamma(T) – \gamma(T) = t(n)\).
The signless \(r\)-associated Stirling numbers of the first kind \(d_r(n, k)\) counts the number of permutations of the set \(\{1,2,\ldots,n\}\) that have exactly \(k\) cycles, each of which is of length greater than or equal to \(r\), where \(r\)is a fixed positive integer. F. Brenti obtained that the generating polynomials of the numbers \(d_r(n, k)\) have only real zeros. Here we consider the location of zeros of these polynomials.
A kite-design of order \(n\) is a decomposition of the complete graph \(K_n\) into kites. Such systems exist precisely when \(n \equiv 0,1 \pmod{8}\). Two kite systems \((X,\mathcal{K}_1)\) and \((X,\mathcal{K}_2)\) are said to intersect in \(m\) pairwise disjoint blocks if \(|\mathcal{K}_1 \cap \mathcal{K}_2| = m\) and all blocks in \(\mathcal{K}_1 \cap \mathcal{K}_2\) are pairwise disjoint. In this paper, we determine all the possible values of \(m\) such that there are two kite-designs of order \(n\) intersecting in \(m\) pairwise disjoint blocks, for all \(n \equiv 0,1 \pmod{8}\).
In this note, we present some upper bounds for the \(k\)th largest eigenvalue of the adjacency matrix as well as the Laplacian matrix of graphs. Special attention is paid to the Laplacian matrix of trees.
Let \(P(G, \lambda)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). A graph \(G\) is chromatically unique, written \(x\)-unique, if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In this paper, we prove that the graph \(\theta(a_1, a_2, \ldots, a_6)\) is \(x\)-unique for exactly two distinct values of \(a_1, a_2, \ldots, a_6\).
In this paper, we give an explicit expression of the genus distributions of \(M_j^n\), for \(j = 1, 2, \ldots, 11\), which are introduced in the previous paper “Orientable embedding distributions by genus for certain types of non-planar graphs”. For a connected graph \(G = (V, E)\) with a cycle, let \(e\) be an edge on a cycle. By adding \(2n\) vertices \(u_1, u_2,u_3 \ldots, u_n, v_1, v_2,v_3 \ldots, v_n\) on \(e\) in sequence and connecting \(u_k, v_k\) for \(1 \leq k \leq n\), a non-planar graph \(G_n\) is obtained for \(n \geq 3\). Thus, the orientable embedding distribution of \(G_n\) by genus is obtained via the genus distributions of \(M_j^n\).
A graph \(G\) is \(N^m\)-locally connected if for every vertex \(v\) in \(G\), the vertices not equal to \(v\) and with distance at most \(m\) to \(v\) induce a connected subgraph in \(G\). In this note, we first present a counterexample to the conjecture that every \(3\)-connected, \(N^2\)-locally connected claw-free graph is hamiltonian and then show that both connected \(N^2\)-locally connected claw-free graph and connected \(N^3\)-locally connected claw-free graph with minimum degree at least three have connected even \([2, 4]\)-factors.
In J.-P. Serre’s \(Lettre \;à\; M. Tsfasman\) \([3]\), an interesting bound for the maximal number of points on a hypersurface of the \(n\)-dimensional projective space \(PG(n,q)\) over the Galois field \(GF(q)\) with \(q\) elements is given. Using essentially the same combinatorial technique as in \([3]\), we provide a bound which is relative to the maximal dimension of a subspace of \(PG(n,q)\) which is completely contained in the hypersurface. The lower that dimension, the better the bound. Next, by using a different argument, we derive a bound which is again relative to the maximal dimension of a subspace of \(PG(n, q)\) which is completely contained in the hypersurface, If that dimension increases for the latter case, the bound gets better.
As such, the bounds are complementary.
In this paper, it is shown that a variation of banana trees is odd graceful, and it is also proved that the variation of banana is graceful and \(\hat{p}\)-labeling in some cases.
In this paper, we consider the generalized Fibonacci and Pell Sequences and then show the relationships between the generalized Fibonacci and Pell sequences, and the Hessenberg permanents and determinants.
In this paper, a sequence representation of Dyck paths is presented, which yields a sequence representation of the Dyck path poset \({D}\) ordered by pattern containment. This representation makes it clear that the Dyck path poset \({D}\) takes the composition poset investigated by Sagan and Vatter as a subposet, and that the pattern containment order on Dyck paths exactly agrees with a generalized subword order also presented by Sagan and Vatter. As applications of the representation, we describe the Möbius function of \({D}\) and establish the Möbius inverse of the rank function of \({D}\) in terms of Dyck sequences. In the end, a Sperner and unimodal subposet of \({D}\) is given.
A graph is said to be determined by its adjacency spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency spectrum. Although all connected graphs of index less than \(2\) are known to be determined by their adjacency spectra, the classification of DS graphs of index less than \(2\) is not complete yet. The purpose of this paper is to characterize all DS graphs of index less than \(2\) with no \(Z_n\) as a component.
Let \(B\) be a bipartite graph. We obtain two new results as follows:(1) Suppose that \(u \in V(B)\) is a vertex such that \(N_B(u)\) contains at least \(|N_B(u)| – 1\) odd vertices. Let \(f : V(B) \to \mathbb{N}\) be the function such that \(f(u) = 1\) and \(f(v) = \lceil d_B(v)/2 \rceil + 1\) for \(v \in V(B) \setminus u\). Then \(B\) is \(f\)-choosable.(2) Suppose that \(u \in V(B)\) is a vertex such that every vertex in \(N_B(u)\) is odd, and \(v \in V(B)\) is an odd vertex that is not adjacent to \(u\). Let \(f : V(B) \to \mathbb{N}\) be the function such that \(f(u) = 1\), \(f(v) = \lceil d_B(v)/2 \rceil\), and \(f(w) = \lceil d_B(w)/2 \rceil + 1\) for \(w \in V(B) \setminus \{u, v\}\). Then \(B\) is \(f\)-choosable.
Assume that \(G = (V, E)\) is an undirected and connected graph, and consider \(C \subseteq V\). For every \(v \in V\), let \(I_r(v) = \{u \in C: d(u,v) \leq r\}\), where \(d(u,v)\) denotes the number of edges on any shortest path between \(u\) to \(v\) in \(G\). If all the sets \(I_r(v)\) for \(v \in V\) are pairwise different, and none of them is the empty set, \(C\) is called an \(r\)-identifying code. In this paper, we consider \(t\)-vertex-robust \(r\)-identifying codes of level \(s\), that is, \(r\)-identifying codes such that they cover every vertex at least \(s\) times and the code is vertex-robust in the sense that \(|I_r(u) \Delta I_r(v)| \geq 2t+1\) for any two different vertices \(u\) and \(v\). Vertex-robust identifying codes of different levels are examined, in particular, of level \(3\). We give bounds (sometimes exact values) on the density or cardinality of the codes in binary hypercubes and in some infinite grids.
A clique \(C\) is an extreme clique of an interval graph \(G\) if there exists some interval model of \(G\) in which \(C\) is the first clique. A graph \(G\) is homogeneously clique-representable if all cliques of \(G\) are extreme cliques. In this paper, we present characterizations of extreme cliques and homogeneously clique-representable graphs.
In this note, we show that there is no \((945, 177, 33)\)-difference set in any group \(G\) of order \(945\) with a normal subgroup \(K\) such that \(G/K \cong \mathbb{C}_{27} \times \mathbb{C}_5\), and hence no cyclic difference set with such parameters exists. This fills one entry of Baumert and Gordon’s table with “No”.
The study of patterns in permutations is a very active area of current research. Klazar defined and studied an analogous notion of pattern for set partitions. We continue this work, finding exact formulas for the number of set partitions which avoid certain specific patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us to conclude that the corresponding sequences are P-recursive. Finally, we define a second notion of pattern in a set partition, based on its restricted growth function. Related results are obtained for this new definition.
Let \(G = (V(G), E(G))\) be a graph with \(\delta(G) \geq 1\). A set \(D \subseteq V(G)\) is a paired-dominating set if \(D\) is a dominating set and the induced subgraph \(G[D]\) contains a perfect matching. The paired domination number of \(G\), denoted by \(\gamma_p(G)\), is the minimum cardinality of a paired-dominating set of \(G\). The paired bondage number, denoted by \(b_p(G)\), is the minimum cardinality among all sets of edges \(E’ \subseteq E\) such that \(\delta(G – E’) \geq 1\) and \(\gamma_p(G – E’) > \gamma_p(G)\). For any \(b_p(G)\) edges \(E’ \subseteq E\) with \(\delta(G – E’) \geq 1\), if \(\gamma_p(G – E’) > \gamma_p(G)\), then \(G\) is called uniformly pair-bonded graph. In this paper, we prove that there exists uniformly pair-bonded tree \(T\) with \(b_p(T) = k\) for any positive integer \(k\). Furthermore, we give a constructive characterization of uniformly pair-bonded trees.
A new construction of a B-T unital using Hermitian curves and certain hypersurfaces of \(\text{PG}(3,q^2)\) is presented. Some properties of an algebraic curve containing all points of a B-T unital are also examined.
A construction of optimal quaternary codes from symmetrical Balanced Incomplete Block (BIB) design \((4t – 1, 2t – 1, t – 1)\) is described.
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, we derive new lower bounds for the Ramsey numbers: \(R(6,8) \geq 129\), \(R(7,9) \geq 235\) and \(R(8,17) \geq 937\). The new bounds are obtained with a constructive method proposed by Xu and Xie et al. and the help of computer algorithm.
We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of face-boundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broad-based approach to calculating genus distributions for graphs synthesized from smaller graphs.
The super (resp., edge-) connectivity of a connected graph is the minimum cardinality of a vertex-cut (resp., an edge-cut) whose removal does not isolate a vertex. In this paper, we consider the two parameters for a special class of graphs \(G(G_p,G_1; M)\), proposed by Chen et al [Applied Math. and Computation, \(140 (2003), 245-254]\), obtained from two \(k\)-regular \(k\)-connected graphs \(G_p\) and \(G_1\), with the same order by adding a perfect matching between their vertices. Our results improve ones of Chen et al. As applications, the super connectivity and the super edge-connectivity of the \(n\)-dimensional hypercube, twisted cube, cross cube, Möbius cube and locally twisted cube are all \(2n – 2\).
We investigate the existence of \(3\)-designs and uniform large sets of \(3\)-designs with block size \(6\) admitting \(\text{PSL}(2, 2^n)\) as an automorphism group.
A simple graph \( G = (V(G), E(G)) \) admits an \( H \)-covering if every edge in \( E(G) \) belongs to a subgraph of \( G \) that is isomorphic to \( H \). An \((a,d)\)-\( H \)-\({antimagic\; total \;labeling}\) of \( G \) is a bijective function \( \xi : V(G) \cup E(G) \to \{1,2,\dots,|V(G)| + |E(G)|\} \) such that for all subgraphs \( H’ \) isomorphic to \( H \), the \( H \)-weights \( w(H’) = \sum_{v \in V(H’)} \xi(v) + \sum_{e \in E(H’)} \xi(e) \) constitute an arithmetic progression \( a, a+d, a+2d, \dots, a+(t-1)d \), where \( a \) and \( d \) are positive integers and \( t \) is the number of subgraphs of \( G \) isomorphic to \( H \). Additionally, the labeling \( \xi \) is called a \({super}\) \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) if \( \xi(V(G)) = \{1, 2, \dots, |V(G)|\} \).
In this paper, we introduce the notion of \((a, d)\)-\( H \)-\({antimagic\; total\; labeling}\) and study some basic properties of such labeling. We provide an example of a family of graphs obtaining the labelings, that is providing \((a, d)\)-cycle-antimagic labelings of fans.
Let \( j \geq 2 \) be a natural number. For graphs \( G \) and \( H \), the size multipartite Ramsey number \( m_j(G, H) \) is the smallest natural number \( t \) such that any \( 2 \)-coloring by red and blue on the edges of \( K_{j \times t} \) necessarily forces a red \( G \) or a blue \( H \) as a subgraph. Let \( P_n \) be a path on \( n \) vertices. In this note, we determine the exact value of the size multipartite Ramsey number \( m_j(P_4, P_n) \) for \( n \geq 2 \).
In this paper, we determine the Ramsey number for the disjoint union of graphs versus a graph \( H \), especially \( R\left(\bigcup_{i=1}^{k} l_i S_{n_i}(1,1), W_6\right) \), \( R\left(\bigcup_{i=1}^{k} l_i S_{n_i}(1,2), W_6\right) \), and \( R\left(\bigcup_{i=1}^k l_i S_{n_i}, W_4\right) \).
In this paper, we discuss an upper bound for exponents of loopless asymmetric two-colored digraphs. If \( D \) is an asymmetric primitive two-colored digraph on \( n \) vertices, we show that \( \text{exp}(D) \leq 3n^2 + 2n – 2 \). For an asymmetric two-colored digraph \( D \) which contains a primitive two-colored cycle of length \( s \leq n \), we show its exponent is at most \( \frac{s^2 – 1}{2} + (s + 1)(n – s) \). We characterize such two-colored digraphs whose exponents equal \( \frac{s^2 – 1}{2} + (s + 1)(n – s) \) and show that the largest exponent of an asymmetric two-colored digraph lies in the interval \( \left[\frac{n^2 – 1}{2}, 3n^2 + 2n – 2\right] \) when \( n \) is odd, or \( \left[\frac{n^2}{2}, 3n^2 + 2n – 2\right] \) otherwise.
Let \( G = (V(G), E(G)) \) be a simple graph and let \( f \) be a function from \( V(G) \) to a subset of positive integers. An \( f \)-\({coloring}\) of \( G \) is a generalized edge-coloring such that every vertex \( v \in V(G) \) has at most \( f(v) \) edges colored with the same color. The minimum number of colors needed to define an \( f \)-coloring of \( G \) is called the \( f \)-\({chromatic\; index}\) of \( G \), and denoted by \( \chi_f'(G) \). The \( f \)-chromatic index of \( G \) is equal to \( \Delta_f(G) \) or \( \Delta_f(G) + 1 \), where \( \Delta_f(G) = \max \left\{ \frac{d(v)}{f(v)} \mid v \in V(G) \right\} \). \( G \) is called in the \({class-1}\), denoted by \( C_f1 \), if \( \chi_f'(G) = \Delta_f(G) \); otherwise \( G \) is called in the \({class-2}\), denoted by \( C_f2 \). In this paper, we show that the corona product of a cycle with either the complement of a complete graph, or a path, or a cycle is in \( C_f1 \).
For a simple graph \( G \) with the vertex set \( V \) and the edge set \( E \), a labeling \( \lambda: V \cup E \to \{1, 2, 3, \ldots, k\} \) is called a vertex-irregular total \( k \)-labeling of \( G \) if for any two different vertices \( x \) and \( y \) in \( V \) we have \( wt(x) \neq wt(y) \), where \( wt(x) = \lambda(x) + \sum_{xy \in E} \lambda(xy) \).
The total vertex-irregular strength, denoted by \( tvs(G) \), is the smallest positive integer \( k \) for which \( G \) has a vertex-irregular total \( k \)-labeling. In this paper, we determine the total vertex-irregular strength of a disjoint union of \( t \) copies of a path, denoted by \( tP_n \). We prove that for any \( t \geq 2 \),
\[
tvs(tP_n) =
\begin{cases}
t & \text{for } n = 1, \\
t+1 & \text{for } 2 \leq n \leq 3, \\
\lceil \frac{nt+1}{3} \rceil & \text{for } n \geq 4.
\end{cases}
\]
Let \( G = (V, E) \) be a graph with order \(|G|\) and size \(|E|\). An \((a, d)\)-vertex-antimagic total labeling is a bijection \( \alpha \) from the set of all vertices and edges to the set of consecutive integers \(\{1, 2, \ldots, |V| + |E|\}\), such that the weights of the vertices form an arithmetic progression with the initial term \( a \) and the common difference \( d \). If \( \alpha(V(G)) = \{1, 2, \ldots, |V|\} \) then we call the labeling super \((a, d)\)-vertex antimagic total. In this paper, we show some basic properties of such labelings on a disjoint union of regular graphs and demonstrate how to construct such labelings for some classes of graphs, such as cycles, generalized Petersen graphs, and circulant graphs, for \( d = 1 \).
In this paper, we determine the partition dimension of a complete multipartite graph \( K_{n_1, n_2, \ldots, n_r} \), namely \( pd(K_{n_1, n_2, \ldots, n_r}) \). We show that \( pd(K_{n_1, n_2, \ldots, n_r}) = r + n – 1 \) if \( n_i = n \) for \( 1 \leq i \leq r \), and \( pd(K_{n_1, n_2, \ldots, n_r}) = r + n – 2 \) for \( n = n_1 \geq n_2 \geq \ldots \geq n_r \). We also show that the partition dimension of the caterpillar graph \( C_n^m \) is \( m \) for \( n \leq m \) and \( m + 1 \) for \( n > m \), and the partition dimension of the windmill graph \( W_{2}^{n} \) is \( k \), where \( k \) is the smallest integer such that \( \binom{k}{2} \geq m \).
Let \( G \) be a graph with vertex set \( V = V(G) \) and edge set \( E = E(G) \), and let \( n = |V(G)| \) and \( e = |E(G)| \). A \({vertex-magic\; total\; labeling}\) (VMTL) of a graph is defined as a one-to-one mapping taking the vertices and edges onto the set of integers \(\{1, 2, \ldots, n+e\}\), with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex. In this paper, we present the vertex-magic total labeling of the disjoint union of \( t \) generalized Petersen graphs \(\bigcup_{j=1}^t P(n_j, m_j)\), and the disjoint union of \( t \) special circulant graphs \(\bigcup_{j=1}^t C_n(1, m_j)\).
A \((p,q)\)-graph \( G \) is called \((a,d)\)-\({edge\; antimagic\; total}\), in short \((a,d)\)-EAMT, if there exist integers \( a > 0 \), \( d \geq 0 \) and a bijection \( \lambda: V \cup E \to \{1, 2, \ldots, p+q\} \) such that \( W = \{w(xy) : xy \in E\} = \{a, a+d, \ldots, a + (q-1)d\} \), where \( w(xy) = \lambda(x) + \lambda(y) + \lambda(xy) \) is the edge-weight of \( xy \). An \((a,d)\)-EAMT labeling \( \lambda \) of \( G \) is \({super}\), in short \((a,d)\)-SEAMT, if \( \lambda(V) = \{1, 2, \ldots, p\} \). In this paper, we propose some theorems on how to construct new (bigger) \((a, d)\)-SEAMT graphs from old (smaller) ones.
Let \( G = (V, E) \) be a graph with \( V(G) \) as a set of vertices and \( E(G) \) as a set of edges, where \( n = |V(G)| \) and \( e = |E(G)| \). A graph \( G = (V, E) \) is said to be \((a, d)\)-vertex antimagic total if there exist positive integers \( a \), \( d \), and a bijection \( \lambda \) from \( V(G) \cup E(G) \) to the set of consecutive integers \(\{1, 2, \ldots, n+e\}\) such that the weight of vertices forms an arithmetical progression with initial term \( a \) and common difference \( d \). In this paper, we will give \((a, d)\)-vertex antimagic total labeling of disconnected graphs, which consists of the union of \( t \) suns for \( d \in \{1, 2, 3, 4, 6\} \).
For given graphs \( G \) and \( H \), the \({Ramsey\; number}\) \( R(G, H) \) is the least natural number \( n \) such that for every graph \( F \) of order \( n \) the following condition holds: either \( F \) contains \( G \) or the complement of \( F \) contains \( H \). In this paper, we determine the Ramsey number for a disjoint union of paths versus the cocktail party graph.
We study the complexity of the longest common subsequence (LCS) problem from a new perspective. By an indeterminate string (i-string, for short) we mean a sequence \( \tilde{X} = \tilde{X}[1]\tilde{X}[2]\ldots \tilde{X}[n] \), where \( \tilde{X}[i] \subseteq \Sigma \) for each \( i \), and \( \Sigma \) is a given alphabet of potentially large size. A subsequence of \( \tilde{X} \) is any usual string over \( \Sigma \) which is an element of the finite (but usually of exponential size) language \( \tilde{X}[i_1]\tilde{X}[i_2]\ldots \tilde{X}[i_p] \), where \( 1 \leq i_1 < i_2 < i_3 \ldots < i_p \leq n \), \( p \geq 0 \). Similarly, we define a supersequence of \( x \). Our first version of the LCS problem is Problem ILCS: for given i-strings \( \tilde{X} \) and \( \tilde{Y} \), find their longest common subsequence. From the complexity point of view, new parameters of the input correspond to \( |\Sigma| \), and maximum size \( \ell \) of the subsets in \( \tilde{X} \) and \( \tilde{Y} \). There is also a third parameter \( \mathcal{R} \), which gives a measure of similarity between \( \tilde{X} \) and \( \tilde{Y} \). The smaller the \( \mathcal{R} \), the lesser is the time for solving Problem ILCS. Our second version of the LCS problem is Problem CILCS (constrained ILCS): for given i-strings \( \tilde{X} \) and \( \tilde{Y} \) and a plain string \( Z \), find the longest common subsequence of \( \tilde{X} \) and \( \tilde{Y} \) which is, at the same time, a supersequence of \( Z \). In this paper, we present several efficient algorithms to solve both ILCS and CILCS problems. The efficiency in our algorithms is obtained in particular by using an efficient data structure for special types of range maxima queries and fast multiplication of boolean matrices.
We introduce a practical disk-based compressed text index that, when the text is compressible, takes much less space than the suffix array. It provides good I/O times for searching, which in particular improve when the text is compressible. In this aspect our index is unique, as most compressed indexes are slower than their classical counterparts on secondary memory. We analyze our index and show experimentally that it is extremely competitive on compressible texts. As side contributions, we introduce a compressed rank dictionary for secondary memory operating in one I/O access, as well as a simple encoding of sequences that achieves high-order compression and provides constant-time random access, both in main and secondary memory.
This paper introduces an automaton model called a dual position automaton (a dual PA), and then gives a bit-parallel algorithm for generating a dual PA from a regular expression (RE). For any RE \( r \) over an alphabet \( \Sigma \), our translation algorithm generates a dual PA consisting of \( \tilde{m}(\tilde{m} + 1) \) bits in \( O(\tilde{m}\lceil \tilde{m}/w \rceil) \) time and space, where \( w \) is the length of a computer word, \( \tilde{m} = \sum_{a \in \Sigma} m_a \), and \( m_a \) is the number of occurrences of an alphabet symbol \( a \) in \( r \). Furthermore, we give a method to construct a compact DFA representation from a dual PA. This DFA representation requires only \( (\tilde{m} + 1) \sum_{a \in \Sigma} 2^{m_a} \) bits. Finally, we show RE matching algorithms using such a DFA representation.
We investigate the problem of efficient representations of intervals of positive integers in TCAM (Ternary Content Addressable Memory). The integers are encoded by binary strings of the same length \( n \) and a TCAM of width \( n \) is a string-oriented representation of arbitrary sets of \( n \)-bit strings in terms of a collection of simple sets, called rules. Each rule is a concatenation (of length \( m \)) of singleton sets (i.e., single digits \( 0 \) and \( 1 \)) or the set \(\{0,1\}\) denoted by \( * \). We consider a family of \( n \)-bit encodings for integers, called dense-tree encodings, which includes the lexicographic encoding (i.e., standard unsigned binary encoding) and the binary reflected Gray encoding. We provide exact bounds (with respect to \( n \)) on the minimal sizes of TCAMs representing a subset of \( n \)-bit strings corresponding to an interval. Some other issues related to the minimal sizes and number of essential rules of TCAMs are also investigated.
This article develops an efficient combinatorial algorithm based on labeled directed graphs and motivated by applications in data mining for designing multiple classifiers. Our method originates from the standard approach described in [37]. It defines a representation of a multiclass classifier in terms of several binary classifiers. We are using labeled graphs to introduce additional structure on the classifier. Representations of this sort are known to have serious advantages. An important property of these representations is their ability to correct errors of individual binary classifiers and produce correct combined output. For every representation like this we develop a combinatorial algorithm with quadratic running time to compute the largest number of errors of individual binary classifiers which can be corrected by the combined multiple classifier. In addition, we consider the question of optimizing the classifiers of this type and find all optimal representations for these multiple classifiers.
A graph is called supermagic if it admits a labeling of its edges by consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper, we prove that the necessary conditions for an \( r \)-regular supermagic graph of order \( n \) to exist are also sufficient. All proofs are constructive and they are based on finding supermagic labelings of circulant graphs.
A distance magic labeling of a graph of order \( n \) is a bijection \( f: V \to \{1, 2, \dots, n\} \) with the property that there is a positive integer constant \( k \) such that for any vertex \( x \), \( \sum_{y \in N(x)} f(y) = k \), where \( N(x) \) is the set of vertices adjacent to \( x \). In this paper, we prove new results about the distance magicness of graphs that have minimum degree one or two. Moreover, we construct distance magic labeling for an infinite family of non-regular graphs.
Since Moore digraphs do not exist for \( k \neq 1 \) and \( d \neq 1 \), the problem of finding digraphs of out-degree \( d \geq 2 \), diameter \( k \geq 2 \) and order close to the Moore bound becomes an interesting problem. To prove the non-existence of such digraphs or to assist in their construction (if they exist), we first may wish to establish some properties that such digraphs must possess. In this paper, we consider the diregularity of such digraphs. It is easy to show that any digraph with out-degree at most \( d \geq 2 \), diameter \( k \geq 2 \) and order one or two less than the Moore bound must have all vertices of out-degree \( d \). However, establishing the regularity or otherwise of the in-degree of such a digraph is not easy. In this paper, we prove that all digraphs of defect two are either diregular or almost diregular. Additionally, in the case of defect one, we present a new, simpler, and shorter proof that a digraph of defect one must be diregular, and in the case of defect two and for \( d = 2 \) and \( k \geq 3 \), we present an alternative proof that a digraph of defect two must be diregular.
In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree \(d\) and \(d^2 + 1\) vertices), and found that such graphs exist only for \(d = 2, 3, 7\) and possibly \(57\). In 1980, Erdős et al., using eigenvalue analysis, showed that, with the exception of \(C_4\), there are no graphs of diameter 2, maximum degree \(d\) and \(d^2\) vertices. In this paper, we show that graphs of diameter 2, maximum degree \(d\) and \(d^2 – 1\) vertices do not exist for most values of \(d\) with \(d \geq 6\), and conjecture that they do not exist for any \(d \geq 6\).
In \([5]\), a product summation of ordered partition \(f(n,m,r) = \sum{c_1^r + c_2^r + \cdots + c_m^r }\) was defined, where for two given positive integers \(m,r\), the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\). \(f(n,r) = \sum_{i=1}^n f(n,m,r)\) was also defined. Many results on \(f(n,m,r)\) were found. However, few things have been known about \(f(n,r)\). In this paper, we give more details for \(f(n,r)\), including its two recurrences, its explicit formula via an entry of a matrix and its generating function. Unexpectedly, we obtain some interesting combinatorial identities, too.
In this paper, we obtain new general results containing sums of binomial and multinomial with coefficients satisfying a general third order linear recursive relations with indices in arithmetic progression.
The closed neighborhood \(N[e]\) 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 \(\{-1,1\}\). If \(\sum_{e \in N[e]} f(x) \geq 1\) for each \(e \in E(G)\), then \(f\) is called a signed edge dominating function of \(G\). The minimum of the values \(\sum_{e \in E(G)} f(e)\), taken over all signed edge dominating functions \(f\) of \(G\), is called the signed edge domination number of \(G\) and is denoted by \(\gamma’_s(G)\). It has been conjectured that \(\gamma’_s(T) \geq 1\) for every tree \(T\). In this paper we prove that this conjecture is true and then classify all trees \(T\) with \(\gamma’_s(T) = 1,2\) and \(3\).
This article is a contribution to the study of block-transitive automorphism groups of \(2\)-\((v,k,1)\) block designs. Let \(\mathcal{D}\) be a \(2\)-\((v,k,1)\) design admitting a block-transitive, point-primitive but not flag-transitive group \(G\) of automorphisms. Let \(k_1 = (k, v-1)\) and \(q = p^f\) for prime \(p\). In this paper we prove that if \(G\) and \(D\) are as above and \(q > {(2(k_rk-k_r+1)f)^{\frac{1}{4}}}\) then \(G\) does not admit a Chevalley group \(E_7(q)\) as its socle.
A graph \(G\) is called super edge-magic if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1, 2, \ldots, |V(G)| + |E(G)|\}\) such that \(f(u) + f(v) + f(uv) = C\) is a constant for any \(uv \in E(G)\) and \(f(V(G)) = \{1, 2, \ldots, |V(G)|\}\), \(f(E(G)) = \{|V(G)| + 1, |V(G)| + 2, \ldots, |V(G)| + |E(G)|\}\). R. M. Figueroa-Centeno et al. provided the following conjecture: For every integer \(n \geq 5\), the book \(B_n\) is super edge-magic if and only if \(n\) is even or \(n \equiv 5 \pmod 8\). In this paper, we show that \(B_n\) is super edge-magic for even \(n \geq 6\).
It was conjectured in \([10]\) that the upper bound for the strong chromatic index \(s'(G)\) of bipartite graphs is \(\Delta(G)^2+1\), where \(\Delta(G)\) is the largest degree of vertices in \(G\). In this note we study the strong edge coloring of some classes of bipartite graphs that belong to the class of partial cubes. We introduce the concept of \(\Theta\)-graph \(\Theta(G)\) of a partial cube \(G\), and show that \(s'(G) \leq \chi(\Theta(G))\) for every tree-like partial cube \(G\). As an application of this bound we derive that \(s'(G) \leq 2\Delta(G)\) if \(G\) is a \(p\)-expansion graph.
We introduce notions of \(k\)-chromatic uniqueness and \(k\)-chromatic equivalence in the class of all Sperner hypergraphs. They generalize the chromatic uniqueness and equivalence defined in the class of all graphs \([10]\) and hypergraphs \([2, 4, 8]\). Using some known facts, concerning a \(k\)-chromatic polynomial of a hypergraph \([5]\), a set of hypergraphs whose elements are \(3\)-chromatically unique is indicated. A set of hypergraphs characterized by a described \(3\)-chromatic polynomial is also shown. The application of the investigated notions can be found in \([5]\).
A graph-pair of order \(t\) is two non-isomorphic graphs \(G\) and \(H\) on \(t\) non-isolated vertices for which \(G \cup H \cong K_t\) for some integer \(t \geq 4\). Given a graph-pair \((G, H)\), we say \((G, H)\) divides some graph \(K\) if the edges of \(K\) can be partitioned into copies of \(G\) and \(H\) with at least one copy of \(G\) and at least one copy of \(H\). We will refer to this partition as a \((G, H)\)-multidecomposition of \(K\).
Let \(V\) denote the \(n\)-dimensional row vector space over a finite field \(\mathbb{F}_q\), and let \(W\) be a subspace of dimension \(n-d\). Let \(L(n,d) = \mathcal{P} \cup \{0\}\), where \({P} = \{A | A \text{ is a subspace of } V, A + W = V\}\). Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.
A graph \(G\) is called \(H\)-equipackable if every maximal \(H\)-packing in \(G\) is also a maximum \(H\)-packing in \(G\). All \(M_2\)-equipackable graphs and \(P_3\)-equipackable graphs have been characterized. In this paper, \(P_k\)-equipackable paths, \(P_k\)-equipackable cycles, \(M_3\)-equipackable paths and \(M_3\)-equipackable cycles are characterized.
Let \(G\) be a graph with \(r\) vertices of degree at least two. Let \(H\) be any graph. Consider \(r\) copies of \(H\). Then \(G \oplus H\) denotes the graph obtained by merging the chosen vertex of each copy of \(H\) with every vertex of degree at least two of \(G\). Let \(T_0\) and \(T^{A_1}\) be any two caterpillars. Define the first attachment tree \(T_1 = T_0 \oplus T^{A_1}\). For \(i \geq 2\), define recursively the \((i^{th})\) attachment tree \(T_i = T_{i-1} \oplus T^{A_i}\), where \(T_{i-1}\) is the \((i-1)^{th}\) attachment tree. Here one of the penultimate vertices of \(T^{A_1}\), \(i \geq 1\) is chosen for merging with the vertices of degree at least two of \(T_{i-1}\), for \(i \geq 1\). In this paper, we prove that for every \(i \geq 1\), the \(i\)th attachment tree \(T_i\) is graceful and admits a \(\beta\)-valuation. Thus it follows that the famous graceful tree conjecture is true for this infinite class of \((i^{th})\) attachment trees \(T’_is\), for all \(i \geq 1\). Due to the results of Rosa \([21]\) and El-Zanati et al. \([5]\) the complete graphs \(K_{2cm+1}\) and complete bipartite graphs \(K_{qm,pm}\), for \(c,p,m,q \geq 1\) can be decomposed into copies of \(i\)th attachment tree \(T_i\), for all \(i \geq 1\), where \(m\) is the size of such \(i\)th attachment tree \(T_i\).
A packing of \(K_n\) with copies of \(C_4\) (the cycle of length \(4\)), is an ordered triple \((V, \mathcal{C}, L)\), where \(V\) is the vertex set of the complete graph \(K_n\), \(C\) is a collection of edge-disjoint copies of \(C_4\), and \(L\) is the set of edges not belonging to a block of \(\mathcal{C}\). The number \(n\) is called the order of the packing and the set of unused edges \(L\) is called the leave. If \(C\) is as large as possible, then \((V, \mathcal{C}, L)\) is called a maximum packing MPC\((n, 4, 1)\). We say that an handcuffed design \(H(v, k, 1)\) \((W, P)\) is embedded into an MPC\((n, 4, 1)\) \((V, C, L)\) if \(W \subseteq V\) and there is an injective mapping \(f : \mathcal{P} \to \mathcal{C}\) such that \(P\) is a subgraph of \(f(P)\) for every \(P \in \mathcal{P}\). Let \(\mathcal{SH}(n, 4, k)\) denote the set of the integers \(v\) such that there exists an MPC\((n, 4, 1)\) which embeds an \(H(v, k, 1)\). If \(n \equiv 1 \pmod 8\) then an MPC\((n, 4, 1)\) coincides with a \(4\)-cycle system of order \(n\) and \(\mathcal{SH}(n, 4, k)\) is found by Milici and Quattrocchi, Discrete Math., \(174 (1997)\).
The aim of the present paper is to determine \(\mathcal{SH}(n, 4, k)\) for every integer \(n \not\equiv 1 \pmod 8\), \(n \geq 4\).
Given a sequence \(X = (x_1, x_2, \ldots, x_k)\), let \(Y = (y_1, y_2, \ldots, y_k)\) be a sequence obtained by rearranging the terms of \(X\). The total self-variation of \(Y\) relative to \(X\) is \(\zeta_X(Y) = \sum_{i=1}^k |y_i – x_i|\). On the other hand, let \(G = (V, E)\) be a connected graph and \(\phi\) be a permutation of \(V\). The total relative displacement of \(\phi\) is \(\delta_\phi(G) = \sum_{\{x \neq y\}\subset V} |d(x, y) – d(\phi(x), \phi(y))|\), where \(d(v, w)\) means the distance between \(v\) and \(w\) in \(G\). It’s clear that the total relative displacement of \(\phi\) is a total self-variation relative to the distance sequence of the graph.
In this paper, we determine the sequences which attain the maximum value of the total self-variation of all possible rearrangements \(Y\) relative to \(X\). Applying this result to the distance sequence of a graph, we find a best possible upper bound for the total relative displacement of a graph.
Let \(G\) be a simple undirected graph. Denote by \(mi(G)\) the number of maximal independent sets in \(G\). In this paper, we determine the second and third largest number of maximal independent sets in trees. Extremal trees achieving these values are also determined.
In \([5]\), the first author posed the problem of determining the spectrum of \((K_4, K_4 – e)\)-designs. In this article, we solve this problem, and also determine the spectrum of \((K_4, K_4 – e)\)-designs with exactly one \(K_4\) (or, equivalently, the spectrum of \((K_4 – e)\)-designs with a hole of size \(4\)). We also improve the bound for embedding a partial \(S(2,4,v)\) into a \((K_4, K_4 – e)\)-design given in \([5]\).
Given a digraph \(D\), its competition graph \(C(D)\) has the same vertex set as \(D\) and an edge between two vertices \(x\) and \(y\) if there is a vertex \(u\) so that \((x,u)\) and \((y,u)\) are arcs of \(D\). Motivated by a problem of communications, Kim and Roberts [2002] studied the competition graphs of the special digraphs known as semiorders and the graphs arising as competition graphs of acyclic digraphs satisfying conditions so called \(C(p)\) or \(C^*(p)\). While they could completely characterize the competition graph of an acyclic digraph satisfying \(C(p)\), they obtained only partial results on \(C^*(p)\) and left the general case open. In this paper, we answer their open question.
A graph \(G\) is said to be well-covered if every maximal independent set of \(G\) is of the same size. It has been shown that characterizing well-covered graphs is a co-NP-complete problem. In an effort to characterize some of these graphs, different subclasses of well-covered graphs have been studied. In this paper, we will introduce the subclass of stable well-covered graphs, which are well-covered graphs that remain well-covered with the addition of any edge. Some properties of stable well-covered graphs are given. In addition, the relationships between stable well-covered graphs and some other subclasses of well-covered graphs, including the surprising equivalence between stable well-covered graphs and other known subclasses, are proved.
Let \(G\) be a graph with vertex set \(V(G)\). For any \(S \subseteq V(G)\), we use \(w(G – S)\) to denote the number of components of \(G – S\). The toughness of \(G\), \(t(G)\), is defined as \(t(G) = \min\left\{\frac{|S|}{w(G – S)} \mid S \subseteq V(G), w(G – S) > 1\right\}\) if \(G\) is not complete; otherwise, set \(t(G) = +\infty\). In this paper, we consider the relationship between the toughness and the existence of fractional \((g, f)\)-factors. It is proved that a graph \(G\) has a fractional \((g, f)\)-factor if \(t(G) \geq \frac{b^2 – 1}{a}\).
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 \(K_{2nt+1}\) (i.e., cyclic \((K_{2nt+1}, G)\)-designs) 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 \(C_m\) by adding an edge joining distinct vertices in the same part in the bipartition of \(V(C_{2m})\) has a \(\gamma\)-labeling if and only if \(m \geq 3\). This, along with results of Blinco and of Froncek, shows that if \(G\) is a graph of size \(n\) consisting of a cycle with a chord, then there exists a cyclic \((K_{2nt+1},G)\)-design for every positive integer \(t\).
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 is a realization of \(\pi\) containing \(H\) as a subgraph. In this paper, we characterize potentially \(K_{1,1,6}\)-positive graphic sequences. This characterization implies the value of \(\sigma(K_{1,1,6}, n)\). Moreover, we also give a simple sufficient condition for a positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) to be potentially \(K_{1,1,s}\)-graphic for \(n \geq s+2\) and \(s \geq 2\).
Let \(H(B)\) denote the space of all holomorphic functions on the unit ball \(B\). Let \(u \in H(B)\) and \(\varphi\) be a holomorphic self-map of \(B\). This paper characterizes the boundedness and compactness of the weighted composition operator \(uC_{\varphi}\), from Bloch-type spaces to weighted-type spaces in the unit ball.
Let \(G\) be a graph of order \(n\). Let \(a\) and \(b\) be integers with \(1 \leq a < b\), and let \(k \geq 2\) be a positive integer not larger than the independence number of \(G\). Let \(g(x)\) and \(f(x)\) be two non-negative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \frac{(a+b)(k(a+b)-2)}{a+1}\) and \(|N_G(x_1) \cup N_G(x_2) \cup \cdots \cup N_G(x_k)| \geq \frac{(b-1)n}{a+b}\) for any independent subset \(\{x_1, x_2, \ldots, x_k\}\) of \(V(G)\). Furthermore, we show that the result is best possible in some sense.
A graph \(G\) is called quasi-claw-free if it satisfies the property:\(d(x,y) = 2 \Rightarrow \text{there exists} u \in N(x) \cap N(y) \text{ such that } N[u] \subseteq N[x] \cup N[y].\) It is shown that a Hamiltonian cycle can be found in polynomial time in four subfamilies of quasi-claw-free graphs.
We study near hexagons which satisfy the following properties:(i) every two points at distance 2 from each other are contained in a unique quad of order \((s,r_1)\) or \((s,r_2), r_1\neq r_2\); (ii) every line is contained in the same number of quads; (iii) every two opposite points are connected by the same number of geodesics. We show that there exists an association scheme on the point set of such a near hexagon and calculate the intersection numbers. We also show how the eigenvalues of the collinearity matrix and their corresponding multiplicities can be calculated. The fact that all multiplicities and intersection numbers are nonnegative integers gives restrictions on the parameters of the near hexagon. We apply this to the special case in which the near hexagon has big quads.
A perfect \(r\)-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance \(r\) of exactly one vertex in the subset. We determine the relationship between perfect \(r\)-codes in the lexicographic product of two simple graphs and perfect \(r\)-codes in each of the factors.
A graph \(G\) is called uniquely \(k\)-list colorable, or \(UkLC\) for short, if it admits a \(k\)-list assignment \(L\) such that \(G\) has a unique \(L\)-coloring. A graph \(G\) is said to have the property \(M(k)\) (\(M\) for Marshal Hall) if and only if it is not \(UkLC\). The \(m\)-number of a graph \(G\), denoted by \(m(G)\), is defined to be the least integer \(k\) such that \(G\) has the property \(M(k)\). After M. Mahdian and E.S. Mahmoodian characterized the \(U2LC\) graphs, M. Ghebleh and E.S. Mahmoodian characterized the \(U3LC\) graphs for complete multipartite graphs except for nine graphs in 2001. Recently, W. He et al. verified all the nine graphs are not \(U3LC\) graphs. Namely, the \(U3LC\) complete multipartite graphs are completely characterized. In this paper, complete multipartite graphs whose \(m\)-number are equal to \(4\) are researched and the \(U4LC\) complete multipartite graphs, which have at least \(6\) parts, are characterized except for finitely many of them. At the same time, we give some results about some complete multipartite graphs whose number of parts is smaller than \(6\).
A list-assignment \(L\) to the vertices of \(G\) is an assignment of a set \(L(v)\) of colors to vertex \(v\) for every \(v \in V(G)\). An \((L,d)^*\)-coloring is a mapping \(\phi\) that assigns a color \(\phi(v) \in L(v)\) to each vertex \(v \in V(G)\) such that at most \(d\) neighbors of \(v\) receive color \(\phi(v)\). A graph is called \((k,d)^*\)-choosable, if \(G\) admits an \((L,d)^*\)-coloring for every list assignment \(L\) with \(|L(v)| \geq k\) for all \(v \in V(G)\). In this note, it is proved that:(1) every toroidal graph containing neither adjacent \(3\)-cycles nor \(5\)-cycles, is \((3,2)^*\)-choosable;(2) every toroidal graph without \(3\)-cycles, is \((3,2)^*\)-choosable.
In this note, we consider a generalized Fibonacci sequence \(\{u_n\}\). Then we give a generating matrix for the terms of sequence \(\{u_{kn}\}\) for a positive integer \(k\). With the aid of this matrix, we derive some new combinatorial identities for the sequence \(\{u_{kn}\}\).
Let \(G = (V, E)\) be a graph. A subset \(S\) of \(V\) is called a dominating set of \(G\) if every vertex in \(V – S\) is adjacent to at least one vertex in \(S\). A global dominating set is a subset \(S\) of \(V\) which is a dominating set of both \(G\) as well as its complement \(\overline{G}\). The domination number (global domination number) \(\gamma(\gamma_g)\) of \(G\) is the minimum cardinality of a dominating set (global dominating set) of \(G\). In this paper, we obtain a characterization of bipartite graphs with \(\gamma_g = \gamma + 1\). We also characterize unicyclic graphs and bipartite graphs with \(\gamma_g = \alpha_0(G) + 1\), where \(\alpha_0(G)\) is the vertex covering number of \(G\).
In paper \([7]\), S. J. Xu and W. Jin proved that a cyclic group of order \(pq\), for two different odd primes \(p\) and \(q\), is a \(3\)-BCI-group, and a finite \(p\)-group is a weak \((p – 1)\)-BCI-group. As a continuation of their works, in this paper, we prove that a cyclic group of order \(2p\) is a \(3\)-BCI-group, and a finite \(p\)-group is a \((p – 1)\)-BCI-group.
Fifty-five new or improved lower bounds for \(A(n, d, w)\), the maximum possible number of binary vectors of length \(n\), weight \(w\), and pairwise Hamming distance no less than \(d\), are presented.
We give some estimates of the norm of weighted composition operators from \(\alpha\)-Bloch spaces to Bloch-type spaces on the unit ball in \(7\).
Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\)
if \(G\) is not complete. Otherwise, set \(I(G) = |V(G)| – 1\). Let \(a\) and \(b\) be positive integers such that \(1 \leq a \leq b\), and let \(g(x)\) and \(f(x)\) be positive integral-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\). Let \(h(e) \in [0,1]\) be a function defined on \(E(G)\), and let \(d(x) = \sum_{e \in E_x} h(e)\) where \(E_x = \{xy : y \in V(G)\}\). Then \(d(x)\) is called the fractional degree of \(x\) in \(G\). We call \(h\) an indicator function if \(g(x) \leq d(x) \leq f(x)\) holds for each \(x \in V(G)\). Let \(E^h = \{e : e \in E(G), h(e) \neq 0\}\) and let \(G_h\) be a spanning subgraph of \(G\) such that \(E(G_h) = E^h\). We call \(G_h\) a fractional \((g,f)\)-factor. The main results in this paper are to present some sufficient conditions about isolated toughness for the existence of fractional \((g,f)\)-factors. If \(1 = g(x) < f(x) = b\), this condition can be improved and the improved bound is not only sharp but also a necessary and sufficient condition for a graph to have a fractional \([1,b]\)-factor.
Let \((G,C)\) be an edge-colored bipartite graph with bi-partition \((X,Y)\). A heterochromatic matching of \(G\) is such a matching in which no two edges have the same color. Let \(N^c(S)\) denote a maximum color neighborhood of \(S \subseteq V(G)\).
The spanning tree packing number of a connected graph \(G\), denoted by \(\tau(G)\), is the maximum number of edge-disjoint spanning trees of \(G\). In this paper, we determine the minimum number of edges that must be added to \(G\) so that the resulting graph has spanning tree packing number at least \(k\), for a given value of \(k\).
Let \(\gamma_{\overline{E}}\) and \(\gamma_{\overline{S}}\) be the minus edge domination and minus star domination numbers of a graph, respectively, and let \(\gamma_E\), \(\beta_1\), \(\alpha_1\) be the edge domination, matching, and edge covering numbers of a graph. In this paper, we present some bounds on \(\gamma_{\overline{E}}\) and \(\gamma_{\overline{S}}\) and characterize the extremal graphs of even order \(n\) attaining the upper bound \(\frac{n}{2}\) on \(\gamma_{\overline{E}}\). We also investigate the relationships between the above parameters.
The Wiener index of a connected graph is defined as the sum of all distances between unordered pairs of vertices. We determine the unicyclic graphs of given order, cycle length and number of pendent vertices with minimum Wiener index.
In this paper, by using the generating functions of Fibonacci polynomial sequences and their partial derivatives, we work out some identities involving the Fibonacci polynomials. As their primary applications, we obtain several identities involving the Fibonacci numbers and Lucas numbers.
Fukuda and Handa \([7]\) asked whether every even partial cube \(G\) is harmonic-even. It is shown that the answer is positive if the isometric dimension of \(G\) equals its diameter which is in turn true for partial cubes with isometric dimension at most \(6\). Under an additional technical condition it is proved that an even partial cube \(G\) is harmonic-even or has two adjacent vertices whose diametrical vertices are at distance at least \(4\). Some related open problems are posed.
By means of partial fraction decomposition, the purpose of this paper is to obtain a generalization of an algebraic identity which was given by Chu in \(\textit{The Electronic J. Camb.}\), \(11(2004), \#N15\).
Let \(G\) be a graph on \(n\) vertices \(v_1, v_2, \ldots, v_n\) and let \(d(v_i)\) be the degree of the vertex \(v_i\). If \((d(v_1), d(v_2), \ldots, d(v_n))^t\) is an eigenvector of the \((0,1)\)-adjacency matrix of \(G\), then \(G\) is said to be harmonic. A semi-regular harmonic graph is the harmonic graph which has exactly two different degrees. An equi-bipartite harmonic graph is the bipartite graph \(H = (X, Y; E)\) with \(|X| = |Y|\). In this paper, we characterize the semi-regular harmonic graph and equi-bipartite harmonic graph, and the degree sequence of equi-bipartite \(3\)-harmonic graphs.
We give necessary and sufficient conditions for a resolvable \(4\)-decomposition of \(AK_n\), in the case where \(H\) is one of the 10 graphs obtained by the union of two paths of length 2, with two possible exceptions. In particular, we complete the \(4\)-star (\(\lambda\)) and \(T\) (\(\tau\)) for higher \(\lambda\) and give complete solutions for resolvable decompositions into Fish (\(4\)-\(3\)), Mulinetto (\(hx\)) and Kites (\(BSI\)). In the cases of the Fish and Mulinetto the solution is obtained \(1\)-rotationally.
We note that with only a slight modification, Su’s proof on the fragments in \(k\)-critical \(n\)-connected graphs (see J. Graph Theory \(45 (2004), 281-297\)) can imply the following more general result: every non-complete \(W\)-locally \(k\)-critical \(n\)-connected graph has \(2k + 2\) distinct fragments \(F_1, F_2, \ldots, F_{2k+2}\) such that \(F_1 \cap W, F_2 \cap W, \ldots, F_{2k+2} \cap W\) are pairwise disjoint.
A packing of a graph \(G\) is a set of edge-disjoint \(4\)-cycles in \(G\) and a maximum packing of \(G\) with \(4\)-cycles is a packing which contains the largest number of \(4\)-cycles among all packings of \(G\). In this paper, we obtain the maximum packing of certain graphs such as \(K_{2m+1} – H\) where \(H\) is a \(2\)-regular subgraph, \(K_{2m} – F\) where \(F\) is a spanning odd forest of \(K_{2m}\), and \(2K_{2m} – L\) where \(L\) is a \(2\)-regular subgraph of \(2K_{2m}\).
In this paper, we consider the relationships between the second order linear recurrences, and the generalized doubly stochastic permanents and determinants.
The distance \( d(u, v) \) between a pair of vertices \( u \) and \( v \) in a connected graph \( G \) is the length of a shortest path joining them. A vertex \( v \) of a connected graph \( G \) is an eccentric vertex of a vertex \( u \) if \( v \) is a vertex at greatest distance from \( u \); while \( v \) is an eccentric vertex of \( G \) if \( v \) is an eccentric vertex of some vertex of \( G \). A vertex \( v \) of \( G \) is a boundary vertex of a vertex \( u \) if \( d(u,w) \leq d(u,v) \) for each neighbour \( w \) of \( v \). A vertex \( v \) is a boundary vertex of \( G \) if \( v \) is a boundary vertex of some vertex of \( G \). It is easy to see that for a vertex \( u \), its eccentric vertices are boundary vertices for \( u \); but not conversely. In this paper, we introduce a new type of eccentricity called b-eccentricity and we study its properties.
A labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \((x,y)\) is the absolute value of the difference of the labels of \(x\) and \(y\). We say that a labeling of the vertices of a graph of order \(n\) is minimally \(k\)-equitable if the vertices are labeled with \(1, 2, \dots, n\) and in the induced labeling of its edges, every label either occurs exactly \(k\) times or does not occur at all. In this paper, we prove that Butterfly and Benes networks are minimally \(2^r\)-equitable where \(r\) is the dimension of the networks.
The method of large scale group testing has been used in the economical testing of blood samples, and in non-testing situations such as experimental designs and coding theory, for over \(50\) years. Some very basic questions addressing the minimum number of tests required to identify defective samples still remain unsolved, including the situation where one defective sample in each of two batches are to be found. This gives rise to an intriguing graph theoretical conjecture concerning bipartite graphs, a conjecture which in this paper is proved to be true in the case where vertices in one part of the bipartite graph have low degree.
Using the action of the linear fractional groups \( L_2(q) \), where \( q = 8, 25, 27, 29, 31 \) and \( 32 \), some \( 1 \)-designs are constructed. It is shown that subgroups of the automorphism group of \( L_2(q) \) appear as the full automorphism group of the constructed designs. In the cases \( q = 8 \) and \( 32 \), it is shown that the symmetric groups \( \mathbb{S}_9 \) and \( \mathbb{S}_{33} \), respectively, appear as the automorphism group of one of the constructed designs.
Here we consider string matching problems that arise naturally in applications to music retrieval. The \( \delta \)-Matching problem calculates, for a given text \( T_{1..n} \) and a pattern \( P_{1..m} \) on an alphabet of integers, the list of all indices \( \mathcal{I}_\delta = \{1 \leq i \leq n-m+1 : \max_{j=1}^m \{|P_j – T_{i+j-1}| \leq \delta\}\} \). The \( \gamma \)-Matching problem computes, for given \( T \) and \( P \), the list of all indices \( \mathcal{I}_\gamma = \{1 \leq i \leq n-m+1 : \sum_{j=1}^m |P_j – T_{i+j-1}| \leq \gamma\} \). In this paper, we extend the current result on the different matching problems to handle the presence of “\emph{don’t care}” symbols. We present efficient algorithms that calculate \( \mathcal{I}_\delta \), \( \mathcal{I}_\gamma \), and \( \mathcal{I}_{(\delta,\gamma)} = \mathcal{I}_\delta \cap \mathcal{I}_\gamma \) for pattern \( P \) with occurrences of “don’t cares”.
In 2004, Kim and Nakprasit showed that the chromatic number of \( K_2(9,4) \) is at least 11. In this note we present an 11-coloring for \( K^2(9,4) \) (the square of the Kneser graph \( K(9,4) \)). This proves that the chromatic number of \( K^2(9,4) \) is \(11\).
Let \( N \) and \( Z \) denote respectively the set of all nonnegative integers and the set of all integers. A \((p,q)\)-graph \( G = (V, E) \) is said to be additively \((a,r)\)-geometric if there exists an injective function \( f : V \to Z \) such that \( f^+(E) = \{a, ar, \dots, ar^{q-1}\} \) where \( a, r \in N \), \( r > 1 \), and \( f^+ \) is defined by \( f^+(uv) = f(u) + f(v) \) for all \( uv \in E \). If further \( f(v) \in N \) for all \( v \in V \), then \( G \) is said to be additively \((a,r)^*\)-geometric. In this paper we characterise graphs which are additively geometric and additively \(^*\)-geometric.
In this paper we find ten new weighing matrices of order \(2n\) and weight \(2n – 5\) constructed from two circulants, by forming a conjecture on the locations of the five zeros in a potential solution. Establishing patterns for the locations of zeros in sequences that can be used to construct weighing matrices seems to be a worthwhile path to explore, as it reduces significantly the computational complexity of the problem.
A dominating set \( S \) in a graph \( G \) is said to be perfect if every vertex of \( G \) not in \( S \) is adjacent to just one vertex of \( S \). Given a vertex subset \( S’ \) of a side \( P_m \) of an \( m \times n \) grid graph \( G \), the perfect dominating sets \( S \) in \( G \) with \( S’ = S \cap V(P_m) \) can be determined via an exhaustive algorithm \( \ominus \) of running time \( O(2^{m+n}) \). Extending \( \ominus \) to infinite grid graphs of width \( m – 1 \), periodicity makes the binary decision tree of \( \ominus \) prunable into a finite threaded tree, a closed walk of which yields all such sets \( S \). The graphs induced by the complements of such sets \( S \) can be codified by arrays of ordered pairs of positive integers via \( \ominus \), for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes \( S \) (with just \( 1 \)-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of \( \ominus \), which allows to show that these sets \( S \) are restrictions of only one total perfect code \( S_1 \) in the integer lattice graph \( \Lambda \) of \( \textbf{R}^2 \). Moreover, the complement \( \Lambda – S_1 \) yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in \( \Lambda \) are in \( 1-1 \) correspondence with the doubly infinite \( \{0, 1\} \)-sequences.
Let \( G \) be a graph of size \( n \) with vertex set \( V(G) \) and edge set \( E(G) \). A \( \rho \)-\({labeling}\) of \( G \) is a one-to-one function \( f : V(G) \to \{0,1,\dots,2n\} \) such that \( \{|f(u) – f(v)| : \{u,v\} \in E(G)\} = \{x_1,x_2,\dots,x_n\} \), where for each \( i \in \{1,2,\dots,n\} \) either \( x_i = i \) or \( x_i = 2n+1-i \). Such a labeling of \( G \) yields a cyclic \( G \)-decomposition of \( K_{2n+1} \). It is conjectured by El-Zanati and Vanden Eynden that every 2-regular graph \( G \) admits a \( \rho \)-labeling. We show that the union of up to ten vertex-disjoint \( C_{4x+1} \) admits a \( \rho \)-labeling.
The Hamilton-Waterloo problem in the case of triangle-factors and Hamilton cycles asks for a \(2\)-factorization of \( K_n \), in which each \(2\)-factor is either a Hamilton cycle or a triangle-factor. Necessarily \( n \equiv 3 \pmod{6} \). The case of \( n \equiv 9 \pmod{18} \) was completely solved in 2004 by Horak, Nedela, and Rosa. In this note, we solve the problem when \( n \equiv 3 \pmod{18} \) and there are at least two Hamilton cycles. A companion paper treats the case when there is exactly one Hamilton cycle and \( n \equiv 3 \pmod{6} \).
There exist \( 3 \) near bowtie systems of order \( 7 \), \( 12 \) bowtie systems of order \( 9 \), and \( 1{,}411{,}422 \) balanced bowtie systems of order \( 13 \).
Chessboard separation problems are modifications to classic chessboard problems, such as the \( N \) Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the \( N + k \) Queens Problem, in which \( k \) Pawns and \( N + k \) mutually non-attacking Queens are to be placed on an \( N \)-by-\( N \) chessboard. Results are presented from performance studies examining the efficiency of sequential and parallel programs that count the number of solutions to the \( N + k \) Queens Problem using traditional backtracking and dancing links. The use of Stochastic Local Search for determining the existence of solutions is also presented. In addition, preliminary results are given for a similar problem, the \( N + k \) Amazons.
In this paper, it is shown that the graph obtained by overlapping the cycle \( C_m \) (\( m \geq 3 \)) and the complete bipartite graph \( K_{3,3} \) at an edge is uniquely determined by its chromatic polynomial.
A graph \( G \) is said to be in the collection \( M_t \) if there are precisely \( t \) different sizes of maximal independent sets of vertices in \( G \). For \( G \in M_t \), and \( v \in G \), we determine the extreme values that \( x \) can assume where \( G \setminus \{v\} \) belongs to \( M_x \). For both the minimum and maximum values, graphs are given that achieve them, showing that the bounds are sharp. The effect of deleting an edge from \( G \) on the number of sizes of maximal independent sets is also considered.
The chromatic polynomial of a graph \( G \), \( P(G; \lambda) \), is the polynomial in \( \lambda \) which counts the number of distinct proper vertex \( \lambda \)-colorings of \( G \), given \( \lambda \) colors. We compute \( P(C_4 \times P_n; \lambda) \) and \( P(C_5 \times P_n; \lambda) \) in matrix form and will find the generating function for each of these sequences.
The \( n \)-cube is the graph whose vertices are all binary words of length \( n > 1 \) and whose edges join vertices that differ in exactly one entry; i.e., are at Hamming distance \( 1 \) from each other. If a word has a non-empty prefix, not the entire word, which is also a suffix, then it is said to be bordered. A word that is not bordered is unbordered. Unbordered words have been studied extensively and have applications in synchronizable coding and pattern matching. The neighborhood of an unbordered word \( w \) is the word itself together with the set of words at Hamming distance \( 1 \) from \( w \). Over the binary alphabet, the neighborhood of an unbordered word \( w \) always contains two bordered words obtained by complementing the first and last entries of \( w \). We determine those unbordered words \( w \) whose neighborhoods otherwise contain only unbordered words.
Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A labeling \( f : V \to \{0,1\} \) induces a partial edge labeling \( f^* : E \to \{0,1\} \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{|f^{*-1}(0) – f^{*-1}(1)| : |f^{-1}(0) – f^{-1}(1)| \leq 1\} \). In this paper, we study the balance index sets of graphs which are \( L \)-products with cycles and complete graphs.
For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) between \( u \) and \( v \) is the length of a longest \( u – v \) path in \( G \). The detour diameter \( \text{diam}_D(G) \) of \( G \) is the greatest detour distance between two vertices of \( G \). Two vertices \( u \) and \( v \) are detour antipodal in \( G \) if \( D(u,v) = \text{diam}_D(G) \). The detour antipodal graph \( \text{DA}(G) \) of a connected graph \( G \) has the same vertex set as \( G \) and two vertices \( u \) and \( v \) are adjacent in \( \text{DA}(G) \) if \( u \) and \( v \) are detour antipodal vertices of \( G \). For a connected graph \( G \) and a nonnegative integer \( r \), define \( \text{DA}^r(G) \) as \( G \) if \( r = 0 \) and as the detour antipodal graph of \( \text{DA}^{r-1}(G) \) if \( r > 0 \) and \( \text{DA}^{r-1}(G) \) is connected. Then \( \{\text{DA}^r(G)\} \) is the detour antipodal sequence of \( G \). A graph \( H \) is the limit of \( \{\text{DA}^r(G)\} \) if there exists a positive integer \( N \) such that \( \text{DA}^r(G) \cong H \) for all \( r \geq N \). It is shown that \( \{\text{DA}^r(G)\} \) converges if \( G \) is Hamiltonian. All graphs that are the limit of the detour antipodal sequence of some Hamiltonian graph are determined.
For a vertex \( x \) in a graph \( G \), we define \( \Psi_1(x) \) to be the number of edges in the closed neighborhood of \( x \). Vertex \( x^* \) is a neighborhood champion if \( \Psi_1(x^*) > \Psi_1(x) \) for all \( x \neq x^* \). We also refer to such an \( x^* \) as a unique champion. For \( d \geq 4 \), let \( n_0(1,d) \) be the smallest number such that for every \( n \geq n_0(1,d) \) there exists an \( n \)-vertex \( d \)-regular graph with a unique champion. Our main result is that \( n_0(1,d) \) satisfies \( d+3 \leq n_0(1,d) < 3d+1 \). We also observe that there can be no unique champion vertex when \( d = 3 \).
In this paper, we consider the non-existence of some bi-level orthogonal arrays (O-arrays) of strength six, with \( m \) constraints (\( 6 \leq m \leq 32 \)), and with index set \( \mu \) (\( 1 \leq \mu \leq 512 \)). The results presented here tend to improve upon the results available in the literature.
We present constructions and results about GDDs with two groups and block size five in which each block has configuration \((s, t)\), that is, in which each block has exactly \(s\) points from one of the two groups and \(t\) points from the other. After some results for a general \(k\), \(s\), and \(t\), we consider the \((2,3)\) case for block size \(5\). We give new necessary conditions for this family of GDDs and give minimal or near-minimal index examples for all group sizes \(n \geq 4\) except for \(n = 24s + 17\).
We compute the limiting average connectivity \(\overline{\kappa}\) of the family of \(3\)-regular expander graphs whose members are formed from the finite fields \(\mathbb{Z}_p\), by connecting every \(x \in \mathbb{Z}_p\) with \(x\pm1\) and \(x^{-1}\), all computations performed modulo \(p\). Namely, we show
\[\lim_{p\to\infty} \overline{\kappa}(\mathbb{Z}_p) = 3\]
for primes \(p\). We compare this behavior with an upper bound on the expected value of \(\overline{\kappa}(\mathbb{Z}_n)\) for a more general class \(\{\mathbb{Z}_n\}_{n\in\mathbb{N}}\) of related graphs.
An \(\lambda\)-design on \(v\) points is a set of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks meet in exactly \(\lambda\) points and not all of the blocks have the same size. Ryser’s and Woodall’s \(\lambda\)-design conjecture states that all \(\alpha\)-designs can be obtained from symmetric designs by a complementation procedure. In a previous paper, the author established feasibility criteria for the existence of \(\lambda\)-designs with two block sizes in the form of integrality conditions, equations, inequalities, and Diophantine equations involving various parameters of the designs. In that paper, these criteria and a computer were used to prove that the \(\lambda\)-design conjecture is true for all \(\lambda\)-designs with two block sizes with \(\lambda \leq 90\) and \(\lambda \neq 45\). In this paper, we extend these results and prove that the \(\lambda\)-design conjecture is also true for all \(\lambda\)-designs with two block sizes with \(\lambda = 45\) or \(91 \leq \alpha < 150\).
The binary linear code \(H^\bot_{m,2}\), \(m > 2\), of length \(\binom{m}{2}\) represented by the generator matrix \(H_{m,2}\) consisting of all distinct column strings of length \(m\) and Hamming weight \(2\) is considered. A parity-check matrix \(H^\bot_{m,2}\) is assigned to the code \(H^\bot_{m,2}\). The code \(H_{m,2,3}\), \(m > 3\), of length \(\binom{m}{2} + \binom{m}{3}\) represented by the parity-check matrix \(H_{m,2,3}\) consisting of all distinct column strings of length \(m\) and Hamming weight two or three is also considered. It is shown that \(H^\bot_{m,2}\) and \(H_{m,2,3}\) are optimal stopping redundancy codes, that is for each of these codes the stopping distance of the associated parity-check matrix is equal to the minimum Hamming distance of the code, and the rows of the parity-check matrix are linearly independent. Explicit formulas determining the number of stopping sets of arbitrary size for these codes are given.
For a finite group \(G\) and subsets \(T_1, T_2\) of \(G\), the Bi-Cayley digraph \(D = (V(D), E(D)) = D(G, T_1, T_2)\) of \(G\) with respect to \(T_1\) and \(T_2\) is defined as the bipartite digraph with vertex set \(V(D) = G \times \{0, 1\}\), and for \(g_1, g_2 \in G\), \(((g_1, 0), (g_2, 1)) \in E(D)\) if and only if \(g_2 = t_1 g_1\) for some \(t_1 \in T_1\), and \(((g_1, 1), (g_2, 0)) \in E(D)\) if and only if \(g_1 = t_2 g_2\) for some \(t_2 \in T_2\). If \(|T_1| = |T_2| = k\), then \(D\) is \(k\)-regular. In this paper, the spectra of Bi-Circulant digraphs are determined. In addition, some asymptotic enumeration theorems for the number of directed spanning trees in Bi-Circulant digraphs are presented.
The genus of a graph \(G\), denoted by \(\gamma(G)\), is the minimum genus of an orientable surface in which the graph can be embedded. In the paper, we use the Joint Tree Model to immerse a graph on the plane and obtain an associated polygon of the graph. Along the way, we construct a genus embedding of the edge disjoint union of \(K\) and \(H\), and solve Michael Stiebitz’s proposed conjecture: Let \(G\) be the edge disjoint union of a complete graph \(K\) and an arbitrary graph \(H\). Let \(H’\) be the graph obtained from \(H\) by contracting the set \(V(X)\) to a single vertex. Then
\[\gamma(K) + \gamma(H’) \leq \gamma(G).\]
We investigate brother avoiding round robin doubles tournaments and construct several infinite families. We show that there is a BARRDT(\(x\)) that is not a SAMDRR(\(n\)) for all \(n > 4\).
A digraph \(D(V, E)\) is said to be graceful if there exists an injection \(f: V(G) \to \{0, 1, \ldots, |E|\}\) such that the induced function \(f’: E(G) \to \{1, 2, \ldots, |E|\}\) which is defined by \(f'(u, v) = [f(v) – f(u)] \pmod{|E| + 1}\) for every directed edge \((u, v)\) is a bijection. Here, \(f\) is called a graceful labeling (graceful numbering) of \(D(V, E)\), while \(f’\) is called the induced edge’s graceful labeling of \(D\). In this paper, we discuss the gracefulness of the digraph \(n – \overrightarrow{C}_m\), and prove that \(n – \overrightarrow{C}_m\) is a graceful digraph for \(m = 4, 6, 8, 10\) and even \(n\).
In this note, we consider relative difference sets with the parameter \((m, 2, m-1, \frac{m-2}{2})\) in a group \(G\) relative to a subgroup \(N\). In the splitting case, \(G = H \times N\), we give a lower bound for the size of the commutator group \(H’\), and we show that \(H\) cannot have a homomorphic image which is generalized dihedral. In the non-splitting case, we prove that there is no \((2n, 2, 2n-1, n-1)\) relative difference set in a generalized dihedral group of order \(4n\), \(n > 1\).
Let \(P_n\) be a path with \(n\) vertices. \(P_n^k\), the \(k\)-th power of the path \(P_n\), is a graph on the same vertex set as \(P_n\), and the edges that join all vertices \(x\) and \(y\) if and only if the distance between them is at most \(k\). In this paper, the crossing numbers of \(P_n^k\) are studied. Drawings of \(P_n^k\) are presented and proved to be optimal for the case \(n \leq 8\) and for the case \(k \leq 4\).
A graph is said to be locally grid if the structure around each of its vertices is a \(3 x 3\) grid. As a follow up of the research initiated in \([8]\) and \([9]\) we prove that most locally grid graphs are uniquely determined by their Tutte polynomial.
Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). A graph \(G\) is chromatically unique if for any graph \(H\), \(P(H, \lambda) = P(G, \lambda)\) implies \(H\) is isomorphic to \(G\). In his Ph.D. thesis, Zhao [Theorems 5.4.2 and 5.4.3] proved that for any positive integer \(t \geq 3\), the complete \(t\)-partite graphs \(K(p – k, p, p, \ldots, p)\) with \(p \geq k+2 \geq 4\) and \(K(p-k, p – 1, p, \ldots, p)\) with \(p \geq 2k \geq 4\) are chromatically unique. In this paper, by expanding the technique employed by Zhao, we prove that the complete \(t\)-partite graph \(K(p-k,\underbrace{ p -1, \ldots, p-1}, \underbrace{p, \ldots, p})\) is chromatically unique for integers \(p \geq k+2 \geq 4\) and \(t \geq d+3 \geq 3\).
We present a block diagonalization method for the adjacency matrices of two types of covering graphs. A graph \(Y\) is a covering graph of a base graph \(X\) if there exists an onto graph map \(\pi: Y \to X\) such that for each \(x \in X\) and for each \(y \in \{y \mid \pi(y) = x\}\), the collection of vertices adjacent to \(y\) maps onto the collection of vertices adjacent to \(x \in X\). The block diagonalization method requires the irreducible representations of the Galois group of \(Y\) over \(X\). The first type of covering graph is the Cayley graph over the finite ring \(\mathbb{Z}/p^n\mathbb{Z}\). The second type of covering graph resembles large lattices with vertices \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}\) for large \(n\). For one lattice, the block diagonalization method allows us to obtain explicit formulas for the eigenvalues of its adjacency matrix. We use these formulas to analyze the distribution of its eigenvalues. For another lattice, the block diagonalization method allows us to find non-trivial bounds on its eigenvalues.
Broadcast domination in graphs is a variation of domination in which different integer weights are allowed on vertices and a vertex with weight \(k\) dominates its distance \(k\)-neighborhood. A distribution of weights on vertices of a graph \(G\) is called a dominating broadcast, if every vertex is dominated by some vertex with positive weight. The broadcast domination number \(\gamma_b(G)\) of a graph \(G\) is the minimum weight (the sum of weights over all vertices) of a dominating broadcast of \(G\). In this paper, we prove that for a connected graph \(G\), \(\gamma_b(G) \geq \lceil{2\text{rad}(G)}/{3}\rceil\). This general bound and a newly introduced concept of condensed dominating broadcast are used in obtaining sharp upper bounds for broadcast domination numbers of three standard graph products in terms of broadcast domination numbers of factors. A lower bound for a broadcast domination number of the Cartesian product of graphs is also determined, and graphs that attain it are characterized. Finally, as an application of these results, we determine exact broadcast domination numbers of Hamming graphs and Cartesian products of cycles.
The semigirth \(\gamma\) of a digraph \(D\) is a parameter related to the number of shortest paths in \(D\). In particular, if \(G\) is a graph, the semigirth of the associated symmetric digraph \(G^*\) is \(\ell(G^*) = \lfloor {g(G) – 1}/{2} \rfloor\), where \(g(G)\) is the girth of the graph \(G\). In this paper, some bounds for the minimum number of vertices of a \(k\)-regular digraph \(D\) having girth \(g\) and semigirth \(\ell\), denoted by \(n(k, g; \ell)\), are obtained. Moreover, we construct a family of digraphs which achieve the lower bound for some particular values of the parameters.
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 has been shown that \(\overrightarrow{d}(G \times H) = d(G)\), where \(\times\) denotes the tensor product of graphs, \(H\) is a special type of circulant graph, and the diameter, \(d(G)\), of \(G\) is at least \(4\). Some interesting results have been obtained using this result. Further, it is shown that \(d(P_r \times K_s) = d(P_r)\) for suitable \(r\) and \(s\). Moreover, it is proved that \(\overrightarrow{d}(C_r \times K_s) = d(C_r)\) for appropriate \(r\) and \(s\).
We consider some partitions where even parts appear twice and some where evens do not repeat. Further, we offer a new partition theoretic interpretation of two mock theta functions of order \(8\).
A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. The purpose of this paper is to generalize some known theorems and results of cordial graphs. Specifically, we show that certain combinations of paths, cycles, and stars are cordial.
An edge-magic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1, 2, \ldots, p+q\) with the property that the sum of the labels on an edge and of its endpoints is constant, independent of the choice of edge. The magic strength of a graph \(G\), denoted by \(emt(G)\), is defined as the minimum of all constants over all edge-magic total labelings of \(G\). The maximum magic strength of a graph \(G\), denoted by \(eMt(G)\), is defined as the maximum constant over all edge-magic total labelings of \(G\). A graph \(G\) is called weak magic if \(eMt(G) – emt(G) > p\). In this paper, we study some classes of weak magic graphs.
In the first part of this paper, we present a generalization of complete graph factorizations obtained by labeling the graph vertices by natural numbers. In this generalization, the vertices are labeled by elements of an arbitrary group \(G\), in order to achieve a \(G\)-transitive factorization of the graph.
Vertex colorings of Steiner systems \(S(t,t+1,v)\) are considered in which each block contains at least two vertices of the same color. Necessary conditions for the existence of such colorings with given parameters are determined, and an upper bound of the order \(O(\ln v)\) is found for the maximum number of colors. This bound remains valid for nearly complete partial Steiner systems, too. In striking contrast, systems \(S(t,k,v)\) with \(k \geq t+2\) always admit colorings with at least \(c\cdot v^\alpha\) colors, for some positive constants \(c\) and \(\alpha\), as \(v\to\infty\).
Cwatsets were originally defined as subsets of \(\mathbb{Z}_2^d\) that are “closed with a twist.” Attempts have been made to generalize them, but the generalizations have failed to produce notions of subcwatset and quotient cwatset that behave naturally.
We present a new, abstract definition that appears to avoid these problems. The relationship between this new definition and its predecessor is similar to that between the abstract definition of “group” and its original meaning as a set of permutations. To justify the broader definition, we use small cancellation theory to prove a result analogous to the statement that every group is isomorphic to some permutation group. After developing the notion of a quotient cwatset, we prove an analogue of the First Homomorphism Theorem.
In this paper, we consider a class of recursively defined, full binary trees called Lucas trees and investigate their basic properties. In particular, the distribution of leaves in the trees will be carefully studied. We then go on to show that these trees are \(2\)-splittable, i.e., they can be partitioned into two isomorphic subgraphs. Finally, we investigate the total path length and external path length in these trees, the Fibonacci trees, and other full \(m\)-ary trees.
A tree \(T\) with \(n\) vertices and a perfect matching \(M\) is strongly graceful if \(T\) admits a graceful labeling \(f\) such that \(f(u)+f(v) = n-1\) for every edge \(uv \in M\). Broersma and Hoede \([5]\) conjectured that every tree containing a perfect matching is strongly graceful in \(1999\). We prove that a tree \(T\) with diameter \(D(T) \leq 5\) supports the strongly graceful conjecture on trees. We show several classes of basic seeds and some constructive methods for constructing large scales of strongly graceful trees.
In a previous paper, the first author introduced two classes of generalized Stirling numbers, \(s_m(n,k,p), S_m(n,k,p)\) with \(m = 1\) or \(2\), called \(p\)-Stirling numbers. In this paper, we discuss their determinant properties.
The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the problem of PI index with respect to some simple pericondensed hexagonal systems and we solve it completely.
As a part of the author’s work of enumerating the edge-forwarding indices of Frobenius graphs, I give a class of valency four Frobenius graphs derived from the Frobenius groups \(\mathbb{Z}_{4n^2+1} \rtimes \mathbb{Z}_4\). Following the method of Fang, Li and Praeger, some properties including the diameter and the type of this class of graphs are given (Theorem \(3.2\)).
We address the problem of determining all sets which form minimal covers of maximal cliques for interval graphs. We produce an algorithm enumerating all minimal covers using the C-minimal elements of the interval order, as well as an independence Metropolis sampler. We characterize maximal removable sets, which are the complements of minimal covers, and produce a distinct algorithm to enumerate them. We use this last characterization to provide bounds on the maximum number of minimal covers for an interval order with a given number of maximal cliques, and present some simulation results on the number of minimal covers in different settings.
A directed triple system of order \(v\), denoted by DTS\((v)\), is a pair \((X,\mathcal{B})\) where \(X\) is a \(v\)-set and \(\mathcal{B}\) is a collection of transitive triples on \(X\) such that every ordered pair of \(X\) belongs to exactly one triple of \(\mathcal{B}\). A DTS\((v)\) is called pure and denoted by PDTS\((v)\) if \((x,y,z) \in \mathcal{B}\) implies \((z,y,x) \notin \mathcal{B}\). A large set of disjoint PDTS\((v)\) is denoted by LPDTS\((v)\). In this paper, we establish the existence of LPDTS\((v)\) for \(v \equiv 0,4 \pmod{6}\), \(v\geq 4\).
We extend and give short proofs of some recent results regarding some classes of rational difference equations.
The skewness \(sk(G)\) of a graph \(G = (V, E)\) is the smallest integer \(sk(G) \geq 0\) such that a planar graph can be obtained from \(G\) by the removal of \(sk(G)\) edges. The splitting number \(sp(G)\) of \(G\) is the smallest integer \(sp(G) \geq 0\) such that a planar graph can be obtained from \(G\) by \(sp(G)\) vertex splitting operations. The vertex deletion \(vd(G)\) of \(G\) is the smallest integer \(vd(G) \geq 0\) such that a planar graph can be obtained from \(G\) by the removal of \(vd(G)\) vertices. Regular toroidal meshes are popular topologies for the connection networks of SIMD parallel machines. The best known of these meshes is the rectangular toroidal mesh \(C_m \times C_n\), for which is known the skewness, the splitting number and the vertex deletion. In this work we consider two related families: a triangulation \(T_{m,n}\) of \(C_m \times C_n\) in the torus, and an hexagonal mesh \(H_{m,n}\), the dual of \(\mathcal{T}_{C_m\times C_n}\) in the torus. It is established that \(sp(T_{m,n}) = vd(T_{m,n}) = sk(H_{C_m\times C_n}) = sp(\mathcal{H}_{C_m\times C_n}) = vd(\mathcal{H}_{m,n}) = \min\{m,n\}\) and that \(sk(\mathcal{T}_{C_m\times C_n}) = 2\min\{m, n\}\).
Exploiting the empirical observation that the probability of \(k\) fixed points in a Welch-Costas permutation is approximately the same as in a random permutation of the same order, we propose a stochastic model for the most probable maximal number of fixed points in a Welch-Costas permutation.
Let \(\gamma_c(G)\)be the connected domination number of \(G\) and \(\gamma_t(G)\) be the tree domination number of \(G\). In this paper, we study the connected domination number and tree domination of \(P(n,k)\), and show that \(\gamma_{tr}(P(n, 4)) = \gamma_c(P(n, 4)) = n-1\) for \(n \geq 17\), \(\gamma_{tr}(P(n, 6)) = \gamma_c(P(n, 6)) = n-1\) for \(n \geq 25\), and \(\gamma_{tr}(P(n,8)) = \gamma_c(P(n,8)) = n-1\) for \(n \geq 33\).
A cut \((A, B)\) (where \(B = V – A\)) in a graph \(G = (V, E)\) is called internal if and only if there exists a vertex \(x \in A\) that is not adjacent to any vertex in \(B\) and there exists a vertex \(y \in B\) such that it is not adjacent to any vertex in \(A\). In this paper, we present a theorem regarding the arrangement of cliques in a chordal graph with respect to its internal cuts. Our main result is that given any internal cut \((A, B)\) in a chordal graph \(G\), there exists a clique with \(\kappa(G) + 1\) vertices (where \(\kappa(G)\) is the vertex connectivity of \(G\)) such that it is (approximately) bisected by the cut \((A, B)\). In fact, we give a stronger result: For any internal cut \((A, B)\) of a chordal graph, and for each \(i\), \(0 \leq i \leq \kappa(G) + 1\), there exists a clique \(K_i\) such that \(|A \cap K_i| = \kappa(G) + 1\), \(|A \cap K_i| = i\), and \(|B \cap K_i| = \kappa(G) + 1- i\).
An immediate corollary of the above result is that the number of edges in any internal cut (of a chordal graph) should be \(\Omega(k^2)\) where \(\kappa(G)\). Prompted by this observation, we investigate the size of internal cuts in terms of the vertex connectivity of the chordal graphs. As a corollary, we show that in chordal graphs, if the edge connectivity is strictly less than the minimum degree, then the size of the mincut is at least \(\frac{\kappa(G)(\kappa(G) + 1)}{2}\), where \(\kappa(G)\) denotes the vertex connectivity. In contrast, in a general graph the size of the mincut can be equal to \(\kappa(G)\). This result is tight.
We determine the automorphism group and the spectrum of the folded hypercube. In addition, we define the Bi-folded hypercube and determine its spectrum.
The main result: If the vertices of a connected graph are labelled by positive real numbers such that the number assigned to any vertex is half of the sum of the numbers assigned to the vertices of its neighbourhood, then each label is an integral multiple of the minimum of all labels. Using this, a result proved earlier in [7] is derived: If \(V\) is a linearly dependent subset of a root system in which all roots have the same norm, then one of the roots in \(V\) is an integral combination of the other roots in \(V\).
A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is said to be a dominating set of \(G\) if each \(v \in V – D\) is adjacent to at least one vertex of \(D\). The minimum cardinality of a dominating set of \(G\) is called the domination number of \(G\) and is denoted by \(\gamma(G)\). A dominating set \(D\) with cardinality \(\gamma(G)\) is called a \(\gamma\)-set of \(G\). Given a graph \(G\), a new graph, denoted by \(\gamma.G\) and called the \(\gamma\)-graph of \(G\), is defined as follows: \(V(\gamma.G)\) is the set of all \(\gamma\)-sets of \(G\) and two sets \(D\) and \(S\) of \(V(\gamma.G)\) are adjacent in \(\gamma.G\) if and only if \(|D \cap S| = \gamma(G) – 1\). A graph \(G\) is said to be \(\gamma\)-connected if \(\gamma.G\) is connected. A graph \(G\) is said to be a \(\gamma\)-graph if there exists a graph \(H\) such that \(\gamma-H\) is isomorphic to \(G\). In this paper, we show that trees and unicyclic graphs are \(\gamma\)-graphs. Also, we obtain a family of graphs which are not \(\gamma\)-graphs.
A Cayley graph is a graph constructed out of a group \(\Gamma\) and its generating set \(A\). In this paper, we determine the independent domination number, perfect domination number, and independent dominating sets of \(Cay(\mathbb{Z}_n, A)\), for a specified generating set \(A\) of \(\mathbb{Z}_n\).
In this paper, we introduce an online tessellation partial automaton to recognize partial array languages. We also introduce two classes of partial array languages. We also introduce two classes of partial array languages viz, Local Partial Array Languages (PAL-LOC) Recognizable Partial Array Languages (PAL-REC) and prove PAL-REC is exactly the family of partial array languages recognizable by online tessellation partial automaton.
For a connected graph \(G\) of order \(p \geq 2\), a set \(S \subseteq V(G)\) is a geodetic set of \(G\) if each vertex \(v \in V(G)\) lies on an \(x\)-\(y\) geodesic for some elements \(x\) and \(y\) in \(S\). The minimum cardinality of a geodetic set of \(G\) is defined as the geodetic number of \(G\), denoted by \(g(G)\). A geodetic set of cardinality \(g(G)\) is called a \(g\)-set of \(G\). A connected geodetic set of \(G\) is a geodetic set \(S\) such that the subgraph \(G[S]\) induced by \(S\) is connected. The minimum cardinality of a connected geodetic set of \(G\) is the connected geodetic number of \(G\) and is denoted by \(g_c(G)\). A connected geodetic set of cardinality \(g_c(G)\) is called a \(g_c\)-set of \(G\). Connected graphs of order \(p\) with connected geodetic number \(2\) or \(p\) are characterized. It is shown that for positive integers \(r,d\) and \(n \geq d+1\) with \(r \leq d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(g_c(G) = n\). Also, for integers \(p,d\) and \(n\) with \(2 \leq d \leq p-1\), \(d+1 \leq n \leq p\), there exists a connected graph \(G\) of order \(p\), diameter \(d\) and \(g_c(G) = n\).
For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the \({detour\; distance}\) \(D(u, v)\) is the length of a longest \(u\)-\(v\) path in \(G\). A \(u\)-\(v\) path of length \(D(u,v)\) is called a \(u\)-\(v\) detour. A set \(S \subseteq V\) is called a \({detour \;set}\) of \(G\) if every vertex in \(G\) lies on a detour joining a pair of vertices of \(S\). The \({detour \;number}\) \(dn(G)\) of \(G\) is the minimum order of its detour sets, and any detour set of order \(dn(G)\) is a detour basis of \(G\). A set \(S \subseteq V\) is called a \({connected \;detour \;set}\) of \(G\) if \(S\) is a detour set of \(G\) and the subgraph \(G[S]\) induced by \(S\) is connected. The \({connected\; detour\; number}\) \(cdn(G)\) of \(G\) is the minimum order of its connected detour sets, and any connected detour set of order \(cdn(G)\) is called a \({connected\; detour \;basis}\) of \(G\). Certain general properties of these concepts are studied. The connected detour numbers of certain classes of graphs are determined. The relationship of the connected detour number with the detour diameter is discussed, and it is proved that for each triple \(D, k, p\) of integers with \(3 \leq k \leq p-D-1\) and \(D \geq 4\), there is a connected graph \(G\) of order \(p\) with detour diameter \(D\) and \(cdn(G) = k\). A connected detour set \(S\), no proper subset of which is a connected detour set, is a \({minimal\; connected\; detour\; set}\). The \({upper\; connected \;detour\; number}\) \(cdn^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal connected detour set of \(G\). It is shown that for every pair \(a, b\) of integers with \(5 \leq a \leq b\), there is a connected graph \(G\) with \(cdn(G) = a\) and \(cdn^+(G) = b\).
For two vertices \(u\) and \(v\) in a graph \(G = (V, E)\), the \({detour\; distance}\) \(D(u,v)\) is the length of a longest \(u\)-\(v\) path in \(G\). A \(u\)-\(v\) path of length \(D(u, v)\) is called a \(u\)-\(v\) \({detour}\). A set \(S \subseteq V\) is called an \({edge\; detour \;set}\) if every edge in \(G\) lies on a detour joining a pair of vertices of \(S\). The \({edge \;detour\; number}\) \(dn_1(G)\) of \(G\) is the minimum order of its edge detour sets, and any edge detour set of order \(dn_1(G)\) is an \({edge\; detour\; basis}\) of \(G\). A connected graph \(G\) is called an \({edge\; detour\; graph}\) if it has an edge detour set. Certain general properties of these concepts are studied. The edge detour numbers of certain classes of graphs are determined. We show that for each pair of integers \(k\) and \(p\) with \(2 \leq k < p\), there is an edge detour graph \(G\) of order \(p\) with \(dn_1(G) = k\). An edge detour set \(S\), no proper subset of which is an edge detour set, is a \({minimal\; edge \;detour\; set}\). The \({upper\; edge\; detour\; number}\) \(dn_1^+(G)\) of a graph \(G\) is the maximum cardinality of a minimal edge detour set of \(G\). We determine the upper edge detour numbers of certain classes of graphs. We also show that for every pair \(a, b\) of integers with \(2 \leq a \leq 6\), there is an edge detour graph \(G\) with \(dn_1(G) = a\) and \(dn_1^+(G) = b\).
An orthogonal double cover (ODC) of the complete graph \(K_n\) is a collection \(\mathcal{G} = \{G_1,G_2,\ldots,G_n\}\) of \(n\) subgraphs of \(K_n\), such that every edge of \(K_n\) belongs to exactly two of the \(G_i\)’s and every pair of \(G_i\)’s intersect in exactly one edge. If \(G_i \cong G\) for all \(i \in \{1,2,\ldots,n\}\), then \(\mathcal{G}\) is an ODC of \(K_n\) by \(G\). An ODC of \(K_n\) is \({cyclic}\) (CODC) if the cyclic group of order \(n\) is a subgroup of its automorphism group. In this paper, we find CODCs of complete graphs by the complete multipartite graphs \(K_{2,r,s}\), \(K_{1,1,r,s}\), and \(K_{1,1,1,1,r}\).
An \({Edge\; Roman\; dominating\; function}\) of a graph \(G = (V, E)\) is a function \(f’ : E \to \{0,1,2\}\) satisfying the condition that every edge \(x\) for which \(f'(x) = 0\) is adjacent to at least one edge \(y\) for which \(f'(y) = 2\). The \({weight}\) of an Edge Roman dominating function is the value \(f'(E) = \sum_{x\in E} f'(x)\). The minimum weight of an Edge Roman dominating function on a graph \(G\) is called the \({Edge\; Roman\; domination\; number}\) of \(G\). In this paper, we initiate a study of this parameter.
The energy \(E(G)\) of a graph \(G\) is the sum of the absolute values of the eigenvalues of \(G\). Two graphs \(G_1\) and \(G_2\) are said to be equienergetic if \(E(G_1) = E(G_2)\). In this paper, we outline various classes of equienergetic graphs. These results enable the construction of pairs of noncospectral equienergetic graphs of the same order and of the same size.
In this paper, fuzzy finite state automaton with unique membership transition on an input symbol (uffsa) is defined. It is proved and illustrated that for a given fuzzy finite state automaton (ffsa), there exists an equivalent uffsa. Some closure properties of fuzzy regular languages are studied.
A \((G,H)\)-multifactorization of \(\lambda K_m\) is a partition of the edge set of \(\lambda K_m\) into \(G\)-factors and \(H\)-factors with at least one \(G\)-factor and one \(H\)-factor. Atif Abueida and Theresa O’Neil have conjectured that for any integer \(n \geq 3\) and \(m \geq n\), there is a \((G_n, H_n)\)-multidecomposition of \(\lambda K_m\) where \(G_n = K_{1,n-1}\) and \(H_n = C_n\). In this paper, it is shown that the above conjecture is true for \(m=n\) when
For a path \( P_n \) of order \( n \) and for any odd integer \( k \), \( 1 \leq k \leq n – 3 \), Chartrand et al. have given an upper bound for the radio \( k \)-chromatic number of \( P_n \) as \( \frac{k^2+2k+1}{2} \). Here we improve this bound for \( \frac{n-4}{2} \leq k < \frac{2n-5}{3} \) and \( \frac{2n-5}{3} \leq k \leq n-3 \). They are \( \frac{k^2+k+4}{2} \) and \( \frac{k^2+k+2}{2} \), respectively. Also, we improve the lower bound of Kchikech et al. from \( \frac{k^2+3}{2} \) to \( \frac{k^2+5}{2} \) for odd integer \( k \), \( 3 \leq k \leq n-3 \).
In this paper, we obtain a necessary condition for the Skolem gracefulness of the disjoint union of \( k \) signed stars \( K_{1,r_i}, 1 \leq i \leq k \), which we call a \( k \)-signed star \( St(r_1,r_2,\ldots,r_k) \). We also present results on the Skolem gracefulness of the 2-signed star \( St(r_1,r_2) \).
In this paper, a definition of a variation of the standard notion of the line signed graph of a given signed graph is recalled from [14] and some fundamental results linking it to the notions of jump signed graphs [6] and adjacency signed graphs [21], especially with regard to their states of balance, consistency, and compatibility are obtained.
In this paper, we discuss how the addition of a new edge changes the irregularity strength in \( K_{m,m} \) and \( tC_4 \).
The goal of this article is to provide an overview of all the results currently known regarding the connectedness of path graphs. The proofs we present are only those that illustrate the different techniques employed in obtaining the results.
This is an expository paper addressed to readers with a small degree of familiarity with the field of graph theory and its techniques.
This paper presents some new results on permissible degree sets in polygon visibility graphs (PVGs). If the PVG has \( n \) vertices, we say it is an \( n \)-PVG. We also show some canonical construction techniques for PVGs with given degree sets.
In this paper, we present several graph theory related software systems that we have developed. These systems have been used in learning and research. The systems feature drawing and manipulation of graphs as well as execution of graph algorithms. The systems are: JGraph, a Java-based system for creating graphs and running graph algorithms; Colossus, a visibility graph system; Manohar, a system for computing graceful labelings of graphs (with special emphasis on trees); Graph Algorithm Constructor, which allows the creation of graph algorithms by drawing flow diagrams instead of writing source code. We also describe some examples in which the empirical data generated from these systems have allowed us to discover fundamental properties of graphs.
A simple acyclic graphoidal cover of a graph \( G \) is a collection \( \psi \) of paths in \( G \) such that every path in \( \psi \) has at least two vertices, every vertex of \( G \) is an internal vertex of at most one path in \( \psi \), every edge of \( G \) is in exactly one path in \( \psi \), and any two paths in \( \psi \) have at most one vertex in common. The minimum cardinality of a simple acyclic graphoidal cover of \( G \) is called the simple acyclic graphoidal covering number of \( G \) and is denoted by \( \eta_{as}(G) \). A simple acyclic graphoidal cover \( \psi \) of \( G \) with \( |\psi| = \eta_{as}(G) \) is called a minimum simple acyclic graphoidal cover of \( G \). Two minimum simple acyclic graphoidal covers \( \psi_1 \) and \( \psi_2 \) of \( G \) are said to be isomorphic if there exists an automorphism \( \alpha \) of \( G \) such that \( \psi = \{\alpha(P) : P \in \psi_1\} \). In this paper, we characterize trees, unicyclic graphs, and wheels in which any two minimum simple acyclic graphoidal covers are isomorphic.
In this paper, we study the domination number, the global domination number, the cographic domination number, the global cographic domination number, and the independent domination number of all the graph products which are non-complete extended \( p \)-sums (NEPS) of two graphs.
A sum composite labeling of a \((p,q)\) graph \( G = (V,E) \) is an injective function \( f : V(G) \to \{1,2,\dots,2p\} \) such that the function \( f^+ : E(G) \to C \) is also injective, where \( C \) denotes the set of all composite numbers and \( f^+ \) is defined by \( f^+(uv) = f(u) + f(v) \) for all \( uv \in E(G) \). A graph \( G \) is sum composite if there exists a sum composite labeling for \( G \). We give some classes of sum composite graphs and some classes of graphs which are not sum composite. We prove that it is possible to embed any graph \( G \) with a given property \( P \) in a sum composite graph which preserves the property \( P \), where \( P \) is the property of being connected, eulerian, hamiltonian, or planar. We also discuss the NP-completeness of the problem of determining the chromatic number and the clique number of sum composite graphs.
A \((p,q)\)-graph \( G \) is said to be \((k,d)\)-multiplicatively indexable if there exists an injection \( f : V(G) \to \mathbb{N} \) such that \( f^\times(E(G)) = \{k,k+d,\dots,k+(q-1)d\} \), where \( f^\times : E(G) \to \mathbb{N} \) is defined by \( f^\times(uv) = f(u)f(v) \) for every \( uv \in E(G) \). If further \( f(V(G)) = \{1,2,\dots,p\} \), then \( G \) is said to be a \((k,d)\)-strongly multiplicatively indexable graph. In this paper, we initiate a study of graphs that admit such labellings.
Public Key Cryptosystems (PKC) based on formal language theory and semi groups have been of interest and study. A PKC based on free group has been presented in [7]. Subsequently, another PKC using free partially commutative monoids and groups is studied in [1]. In this paper, we propose a PKC for chain cade pictures that uses a finitely presented group for encryption and free group for decryption. Also, we present another PKC for line pictures in the hexagonal grid, which uses a finitely presented group for encryption and finitely presented free partially commutative group for decryption.
Let \( G = (V, E) \) be a connected graph. A subset \( A \) of \( V \) is called an asteroidal set if for any three vertices \( u,v,w \) in \( A \), there exists a \( u \)-\( v \) path in \( G \) that avoids the neighbourhood of \( w \). The asteroidal chromatic number \( \chi_a \) of \( G \) is the minimum order of a partition of \( V \) into asteroidal sets. In this paper we initiate a study of this parameter. We determine the value of \( \chi_a \) for several classes of graphs, obtain sharp bounds, and Nordhaus-Gaddum type results.
A connected graph \(G\) is said to be odd path extendable if for any odd path \(P\) of \(G\), the graph \(G – V(P)\) contains a perfect matching. In this paper, we at first time introduce the concept of odd path extendable graphs. Some simple necessary and sufficient conditions for a graph to be odd path extendable are given. In particular, we show that if a graph is odd path extendable, it is hamiltonian.
In this paper, we give one construction for constructing large harmonious graphs from smaller ones. Subsequently, three families of graphs are introduced and some members of them are shown to be or not to be harmonious.
A graph is called set reconstructible if it is determined uniquely (up to isomorphism) by the set of its vertex-deleted subgraphs. We prove that all graphs are set reconstructible if all \(2\)-connected graphs \(G\) with \(diam(G) = 2\) and all \(2\)-connected graphs \(G\) with \(diam(G) = diam(\overline{G}) = 3\) are set reconstructible.
A function \(f: V(G) \to \{-1,0,1\}\) defined on the vertices of a graph \(G\) is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every \(v \in V\), \(f(N(v)) \geq 1\), where \(N(v)\) consists of every vertex adjacent to \(v\). The weight of a MTDF is the sum of its function values over all vertices. A MTDF \(f\) is minimal if there does not exist a MTDF \(g: V(G) \to \{-1,0,1\}\), \(f \neq g\), for which \(g(v) \leq f(v)\) for every \(v \in V\). The upper minus total domination number, denoted by \(\Gamma^{-}_{t}(G)\), of \(G\) is the maximum weight of a minimal MTDF on \(G\). A function \(f: V(G) \to \{-1,1\}\) defined on the vertices of a graph \(G\) is a signed total dominating function (STDF) if the sum of its function values over any open neighborhood is at least one. The signed total domination number, denoted by \(\gamma^{s}_{t}(G)\), of \(G\) is the minimum weight of a STDF on \(G\). In this paper, we establish an upper bound on \(\Gamma^{-}_{t}(G)\) of the 5-regular graph and characterize the extremal graphs attaining the upper bound. Also, we exhibit an infinite family of cubic graphs in which the difference \(\Gamma^{-}_t(G) – \gamma^{s}_t(G)\) can be made arbitrarily large.
Let \(G\) be a graph with vertex set \(V(G)\). An edge coloring \(C\) of \(G\) is called an edge-cover coloring, if for each color, the edges assigned with it form an edge cover of \(G\). The maximum positive integer \(k\) such that \(G\) has a \(k\)-edge-cover coloring is called the edge cover chromatic index of \(G\) and is denoted by \(\chi’_c(G)\). It is well known that \(\min\{d(v) – \mu(v) : v \in V(G)\} \leq \chi’_c(G) \leq \delta(G)\), where \(\mu(v)\) is the multiplicity of \(v\) and \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’_c(G) = \delta(G)\), \(G\) is called a graph of CI class, otherwise \(G\) is called a graph of CII class. In this paper, we give a new sufficient condition for a nearly bipartite graph to be of CI class.
Though the well-known Vizing’s conjecture is not true for directed graphs in general, we show that it is true when the digraph and its reversal contain an efficient dominating set. In this paper, we investigate the existence of such sets in directed tori and infinite grids. We give a complete characterization of efficient dominating sets in the \(3\)-dimensional case and show the nonexistence of efficient \(d\)-dominating sets in directed tori for any \(d > 1\) and any dimension \(n > 1\).
For every two vertices \(u\) and \(v\) in a graph \(G\), a \(u-v\) geodesic is a shortest path between \(u\) and \(v\). Let \(I(u,v)\) denote the set of all vertices lying on a \(u-v\) geodesic. For a vertex subset \(S\), let \(I_G(S)\) denote the union of all \(I_G(u,v)\) for \(u,v \in S\). The geodetic number \(g(G)\) of a graph \(G\) is the minimum cardinality of a set \(S\) with \(I_G(S) = V(G)\). For a digraph \(D\), there is analogous terminology for the geodetic number \(g(D)\). The geodetic spectrum of a graph \(G\), denoted by \(S(G)\), is the set of geodetic numbers over all orientations of graph \(G\). The lower geodetic number is \(g^-(G) = \min S(G)\) and the upper geodetic number is \(g^+(G) = \max S(G)\). The main purpose of this paper is to investigate lower and upper geodetic numbers of graphs. Our main results in this paper are:
We estimate the essential norm of the weighted composition operator \(uC_{\varphi}\) from the weighted Bergman space \(A^{p}_{\alpha}(\mathbb{B})\) to the weighted space \(H^{\infty}_{\mu}(\mathbb{B})\) on the unit ball \(\mathbb{B}\), when \(p > 1\) and \(\alpha \geq -1\) (for \(\alpha = -1\), \(A^{p}_{\alpha}\) is the Hardy space \(H^{p}(\mathbb{B})\)). We also give a necessary and sufficient condition for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu}(B)\) to be compact, and for the operator \(uC_{\varphi} : A^{p}_{\alpha}(\mathbb{B}) \to H^{\infty}_{\mu,0}(\mathbb{B})\) to be bounded or compact, when \(p > 0\), \(\alpha \geq -1\).
Let \(G = (V,E)\) be a graph. A set \(S \subseteq V\) is called a restrained dominating set of \(G\) if every vertex not in \(S\) is adjacent to a vertex in \(S\) and to a vertex in \(V – S\). The restrained domination number of \(G\), denoted by \(\gamma_r(G)\), is the minimum cardinality of a restrained dominating set of \(G\). In this paper, we establish an upper bound on \(\gamma_r(G)\) for a connected graph \(G\) by the probabilistic method.
Any vertex labeling \(f: V \to \{0,1\}\) of the graph \(G = (V,E)\) induces a partial edge labeling \(f^*: E \to \{0,1\}\) defined by \(f^*(uv) = f(u)\) if and only if \(f(u) = f(v)\). The balance index set of \(G\) is defined as \(\{|f^{*{-1}}(0) – f^{*{-1}}(1)|: |f^{-1}(0) – f^{-1}(1)| \leq 1\}\). In this paper, we first determine the balance index sets of rooted trees of height not exceeding two, thereby completely settling the problem for trees with diameter at most four. Next we show how to extend the technique to rooted trees of any height, which allows us to derive a method for determining the balance index set of any tree.
We show that partial permutation decoding can be used, and give explicit \(s\)-PD-sets in the symmetric group, where \(s\) is less than the full error-correction capability of the code, for some classes of binary codes obtained from the adjacency matrices of the graphs with vertices the \(\binom{n}{3}\) \(3\)-subsets of a set of size \(n\) with adjacency defined by the vertices as \(3\)-sets being adjacent if they have a fixed number of elements in common.
Let \(G\) be a simple connected graph. For a subset \(S\) of \(V(G)\) with \(|S| = 2n + 1\), let \(\alpha_{(2n+1)}(G,S)\) denote the graph obtained from \(G\) by contracting \(S\) to a single vertex. The graph \(\alpha_{(2n+1)}(G, S)\) is also said to be obtained from \(G\) by an \(\alpha_{(2n+1)}\)-contraction. For pairwise disjoint subsets \(S_1, S_2, \ldots, S_{2n}\) of \(V(G)\), let \(\beta_n(G, S_1, S_2, \ldots, S_{2n})\) denote the graph obtained from \(G\) by contracting each \(S_i\) (\(i = 1, 2, \ldots, 2n\)) to a single vertex respectively. The graph \(\beta_{2n}(G, S_1, S_2, \ldots, S_{2n})\) is also said to be obtained from \(G\) by a \(\beta_{2n}\)-contraction. In the present paper, based on \(\alpha_{(2n+1)}\)-contraction and \(\beta_{2}\)-contraction, some new characterizations for \(n\)-extendable bipartite graphs are given.
A graph \(G\) is quasi-claw-free if it satisfies the property: \(d(x, y) = 2 \Rightarrow\) there exists \(u \in N(x) \cap N(y)\) such that \(N[u] \subseteq N[x] \cup N[y]\). In this paper, we prove that the circumference of a \(2\)-connected quasi-claw-free graph \(G\) on \(n\) vertices is at least \(\min\{3\delta + 2, n\}\) or \(G \in \mathcal{F}\), where \(\mathcal{F}\) is a class of nonhamiltonian graphs of connectivity \(2\). Moreover, we prove that if \(n \leq 40\), then \(G\) is hamiltonian or \(G \in \mathcal{F}\).
Let \(K_{n,n}\) denote the complete bipartite graph with \(n\) vertices in each part. In this paper, it is proved that there is no cyclic \(m\)-cycle system of \(K_{n,n}\) for \(m \equiv 2 \pmod{4}\) and \(n \equiv 2 \pmod{4}\). As a consequence, necessary and sufficient conditions are determined for the existence of cyclic \(m\)-cycle systems of \(K_{n,n}\) for all integers \(m \leq 30\).
We examine a design \(\mathcal{D}\) and a binary code \(C\) constructed from a primitive permutation representation of degree \(2025\) of the sporadic simple group \(M^c L\). We prove that \(\text{Aut}(C) = \text{Aut}(\mathcal{D}) = M^c L\) and determine the weight distribution of the code and that of its dual. In Section \(6\) we show that for a word \(w_i\) of weight \(7\), where \(i \in \{848, 896, 912, 972, 1068, 1100, 1232, 1296\}\) the stabilizer \((M^\circ L)_{w_i}\) is a maximal subgroup of \(M^\circ L\). The words of weight \(1024\) split into two orbits \(C_{(1024)_1}\) and \(C_{(1024)_2}\), respectively. For \(w_i \in C_{(1024)_1}\), we prove that \((M^c L)_{w_i}\) is a maximal subgroup of \(M^c L\).
Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v, G, \lambda)\)-PD \(((v, G,\lambda)\)-CD), is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in at most (at least) \(\lambda\) blocks of \(\mathcal{B}\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. In this paper, we have completely determined the packing number and covering number for the graphs with seven points, seven edges and an even cycle.
In this paper, it is shown that there are exactly \(5\) non-isomorphic abstract ovals of order \(9\), all of them projective. The result has been obtained via an exhaustive search, based on the classification of the \(1\)-factorizations of the complete graph with \(10\) vertices.
A graph \(G\) is said to be \(k\)-degenerate if for every induced subgraph \(H\) of \(G\), \(\delta(H) \leq k\). Clearly, planar graphs without \(3\)-cycles are \(3\)-degenerate. Recently, it was proved that planar graphs without \(5\)-cycles or without \(6\)-cycles are also \(3\)-degenerate. And for every \(k = 4\) or \(k \geq 7\), there exist planar graphs of minimum degree \(4\) without \(k\)-cycles. In this paper, it is shown that each \(C_7\)-free plane graph in which any \(3\)-cycle is adjacent to at most one triangle is \(3\)-degenerate. So it is \(4\)-choosable.
This paper investigates the embedding problem for resolvable group divisible designs with block size \(3\). The necessary and sufficient conditions are determined for all \(\lambda \geq 1\).
We provide combinatorial arguments of some relations between classical Stirling numbers of the second kind and two refinements of these numbers gotten by introducing restrictions to the distances among the elements in each block of a finite set partition.
We provide many new edge-magic and vertex-magic total labelings for the cycles \(C_{nk}\), where \(n \geq 3\) and \(k \geq 3\) are both integers and \(n\) is odd. Our techniques are of interest since known labelings for \(C_{k}\) are used in the construction of those for \(C_{nk}\). This provides significant new evidence for a conjecture on the possible magic constants for edge-magic and vertex-magic cycles.
A total dominating set of a graph \(G\) with no isolated vertex is a set \(S\) of vertices of \(G\) such that every vertex is adjacent to a vertex in \(S\). The total domination number of \(G\) is the minimum cardinality of a total dominating set in \(G\). In this paper, we present several upper bounds on the total domination number in terms of the minimum degree, diameter, girth, and order.
We denote by \((p, q)\)-graph \(G\) a graph with \(p\) vertices and \(q\) edges. An edge-magic total (EMT) labeling on a \((p,q)\)-graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow [1,2,\ldots,p+q]\) with the property that, for each edge \(xy\) of \(G\), \(\lambda(x) + \lambda(xy) + \lambda(y) = k\), for a fixed positive integer \(k\). Moreover, \(\lambda\) is a super edge-magic total labeling (SEMT) if it has the property that \(\lambda(V(G)) = \{1, 2,\ldots,p\}\). A \((p,q)\)-graph \(G\) is called EMT (SEMT) if there exists an EMT (SEMT) labeling of \(G\). In this paper, we propose further properties of the SEMT graph. Based on these conditions, we will give new theorems on how to construct new SEMT (bigger) graphs from old (smaller) ones. We also give the SEMT labeling of \(P_n \cup P_{n+m}\) for possible magic constants \(k\) and \(m = 1, 2\),or \(3\).
A Kirkman packing design \(KPD({w, s^*, t^*}, v)\) is a Kirkman packing with maximum possible number of parallel classes, such that each parallel class contains one block of size \(s\), one block of size \(t\) and all other blocks of size \(w\). A \((k, w)\)-threshold scheme is a way of distributing partial information (shadows) to \(w\) participants, so that any \(k\) of them can determine a key easily, but no subset of fewer than \(k\) participants can calculate the key. In this paper, the existence of a \(KPD({3, 4^*, 5^*}, v)\) is established for every \(v \equiv 3 \pmod{6}\) with \(v \geq 51\). As its consequence, some new \((2, w)\)-threshold schemes have been obtained.
In this paper, we mainly define a semidirect product version of the Schützenberger product and also a new two-sided semidirect product construction for arbitrary two monoids. Then, as main results, we present a generating and a relator set for these two products. Additionally, to explain why these products have been defined, we investigate the regularity for the semidirect product version of Schützenberger products and the subgroup separability for this new two-sided semidirect product.
We consider the connected graphs with a unique vertex of maximum degree \(3\). Two subfamilies of such graphs are characterized and ordered completely by their indices. Moreover, a conjecture about the complete ordering of all graphs in this set is proposed.
Let \(G = (V(G), E(G))\) be a simple graph and \(T(G)\) be the set of vertices and edges of \(G\). Let \(C\) be a \(k\)-color set. A (proper) total \(k\)-coloring \(f\) of \(G\) is a function \(f: T(G) \rightarrow C\) such that no adjacent or incident elements of \(T(G)\) receive the same color. For any \(u \in V(G)\), denote \(C(u) = \{f(u)\} \cup \{f(uv) | uv \in E(G)\}\). The total \(k\)-coloring \(f\) of \(G\) is called the adjacent vertex-distinguishing if \(C(u) \neq C(v)\) for any edge \(uv \in E(G)\). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number \(\chi_{at}(G)\) of \(G\). Let \(G\) be a connected graph. If there exists a vertex \(v \in V(G)\) such that \(G – v\) is a tree, then \(G\) is a \(1\)-tree. In this paper, we will determine the adjacent vertex-distinguishing total chromatic number of \(1\)-trees.
In this paper, we extend the study on packing and covering of complete directed graph \(D_t\) with Mendelsohn triples \([6]\). Mainly, the maximum packing of \(D_t-P\) and \(D_t\cup{P}\) with Mendelsohn triples are obtained respectively, where \(P\) is a vertex-disjoint union of directed cycles in \(D_t\).
In the theory of orthogonal arrays, an orthogonal array is called schematic if its rows form an association scheme with respect to Hamming distances. Which orthogonal arrays are schematic orthogonal arrays and how to classify them is an open problem proposed by Hedayat et al. \([12]\). In this paper, we study the Hamming distances of the rows in orthogonal arrays and construct association schemes according to the distances. The paper gives the partial solution of the problem by Hedayat et al. for symmetric and some asymmetric orthogonal arrays of strength two.
The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI index of gated amalgam.
The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we give formulae to calculate the nullity of \(n\)-vertex bicyclic graphs by means of the maximum matching number.
This note calculates the essential norm of a recently introduced integral-type operator from the Hilbert-Bergman weighted space \(A^2_\alpha(\mathbb{B}), \alpha \geq -1\) to a Bloch-type space on the unit ball \(\mathbb{B} \subset \mathbb{C}^n\).
Let \(G\) be a graph and let \(\sigma_k(G)\) be the minimum degree sum of an independent set of \(k\) vertices. For \(S \subseteq V(G)\) with \(|S| \geq k\), let \(\Delta_k(S)\) denote the maximum value among the degree sums of the subset of \(k\) vertices in \(S\). A cycle \(C\) of a graph \(G\) is said to be a dominating cycle if \(V(G \setminus C)\) is an independent set. In \([2]\), Bondy showed that if \(G\) is a \(2\)-connected graph with \(\sigma_3(G) \geq |V(G)| + 2\), then any longest cycle of \(G\) is a dominating cycle. In this paper, we improve it as follows: if \(G\) is a 2-connected graph with \(\Delta_3(S) \geq |V(G)| + 2\) for every independent set \(S\) of order \(\kappa(G) + 1\), then any longest cycle of \(G\) is a dominating cycle.
Let \(B\) be an \(m \times n\) array in which each symbol appears at most \(k\) times. We show that if \(k \leq \frac{n(n-1)}{8(m+n-2)} + 1\) then \(B\) has a transversal.
Let \(T\) be a partially ordered set whose Hasse diagram is a binary tree and let \(T\) possess a unique maximal element \(1_T\). For a natural number \(n\), we compare the number \(A_T^n\) of those chains of length \(n\) in \(T\) that contain \(1_T\) and the number \(B_T^n\) of those chains that do not contain \(1_T\). We show that if the depth of \(T\) is greater or equal to \(2n + [ n \log n ]\) then \(B_T^n > A_T^n\).
The boundedness and compactness of the weighted composition operator from logarithmic Bloch spaces to a class of weighted-type spaces are studied in this paper.
S.M. Lee proposed the conjecture: for any \(n > 1\) and any permutation \(f\) in \(S(n)\), the permutation graph \(P(P_n, f)\) is graceful. For any integer \(n > 1\), we discuss gracefulness of the permutation graphs \(P(P_n, f)\) when \(f = (123), (n-2, n-1, n), (i, i+1), 1 \leq i \leq n-1, (12)(34)\ldots(2m-1, 2m), 1 \leq m \leq \frac{n}{2}\), and give some general results.
A double-loop network (DLN) \(G(N;r,s)\) is a digraph with the vertex set \(V = \{0,1,\ldots, N-1\}\) and the edge set \(E=\{v \to v+r \pmod{N} \text{ and } v \to v+s \pmod{N} | v \in V\}\). Let \(D(N;r,s)\) be the diameter of \(G(N;r,s)\) and let us define \(D(N) = \min\{D(N;r,s) | 1 \leq r < s < N \text{ and } \gcd(N,r,s) = 1\}\), \(D_1(N) = \min\{D(N;1,s) | 1 < s 0\)). Coppersmith proved that there exists an infinite family of \(N\) for which the minimum diameter \(D(N) \geq \sqrt{3N} + c(\log N)^{\frac{1}{4}}\), where \(c\) is a constant.
In this paper, we consider cycle-partition problems which deal with the case when both vertices and edges are specified and we require that they should belong to different cycles. Minimum degree and degree sum conditions are given, which are best possible.
In this paper, we consider the relationships between the second order linear recurrences and the permanents and determinants of tridiagonal matrices.
We correct and improve results from a recent paper by G. Ren and U. Kahler, which characterizes the Bloch, the little Bloch and Besov space of harmonic functions on the unit ball \({B} \subset \mathbb{R}^n\).
Many approaches to drawing graphs in the plane can be formulated and solved as mathematical programming problems. Here, we consider only drawings of a graph where each edge is drawn as a straight-line segment, and we wish to minimize the number of edge crossings over all such drawings. Some formulations of this problem are presented that lead very naturally to other unsolved problems, some solutions, and some new open problems associated with drawing nonplanar graphs in the plane.
In this paper we introduce right angle path and layer of an array. We construct Kolakoski array and study some combinatorial proper-ties of Kolakoski array. Also we obtain recurrence relation for layers and special elements.
An \({eternal \;1-secure}\) set of a graph \(G = (V, E)\) is defined as a set \(S_0 \subseteq V\) that can defend against any sequence of single-vertex attacks by means of single guard shifts along edges of \(G\). That is, for any \(k\) and any sequence \(v_1, v_2, \ldots, v_k\) of vertices, there exists a sequence of guards \(u_1, u_2, \ldots, u_k\) with \(u_i \in S_{i-1}\) and either \(u_i = v_i\) or \(u_iv_i \in E\), such that each set \(S_i = (S_{i-1} -\{u_i\}) \cup \{v_i\}\) is dominating. It follows that each \(S_i\) can be chosen to be an eternal 1-secure set. The \({eternal \;1-security\; number}\), denoted by \(\sigma_1(G)\), is defined as the minimum cardinality of an eternal 1-secure set. This parameter was introduced by Burger et al. [3] using the notation \(\gamma_\infty\). The \({eternal \;m-security}\) number \(\sigma_m(G)\) is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts. A suitable placement of the guards is called an \({eternal\; m-secure}\) set. It was observed that \(\gamma(G) \leq \sigma_m(G) \leq \beta(G)\). In this paper, we obtain specific values of \(\sigma_m(G)\) for certain classes of graphs, namely circulant graphs, generalized Petersen graphs, binary trees, and caterpillars.
Let \( G \) be a simple graph, and let \( p \) be a positive integer. A subset \( D \subseteq V(G) \) is a \( p \)-\({dominating \;set}\) of the graph \( G \) if every vertex \( v \in V(G) – D \) is adjacent to at least \( p \) vertices of \( D \). The \( p \)-\({domination\; number}\) \( \gamma_p(G) \) is the minimum cardinality among the \( p \)-dominating sets of \( G \). Note that the \( 1 \)-domination number \( \gamma_1(G) \) is the usual \({domination\; number}\) \( \gamma(G) \).
In \(1985\), Fink and Jacobson showed that for every graph \(G\) with \(n\) vertices and \(m\) edges the inequality \(y\),\(\gamma_p(G) \geq n — m/p\) holds. In this paper we present a generalization of this theorem and analyze the \(2\)-domination number \(\gamma_2\) in cactus graphs \(G\) with respect on its relation to the matching number \(\alpha_0\) and the number of odd or rather even cycles in \(G\). Further we show that \(\gamma_2(G) \geq \alpha(G)\) for the cactus graphs \(G\) with at most one even cycle and characterize those which
fulfill \(\gamma_2(G) = \alpha(G)\) or rather \(\gamma_2(G) = \alpha(G) +1\).
We introduce a \( k \)-response set as a set of vertices where responders can be placed so that given any set of \( k \) emergencies, these responders can respond, one per emergency, where each responder covers its own vertex and its neighbors. A weak \( k \)-response set does not have to worry about emergencies at the vertices of the set. We define \( R_k \) and \( r_k \) as the minimum cardinality of such sets. We provide bounds on these parameters and discuss connections with domination invariants. For example, for a graph \( G \) of order \( n \) and minimum degree at least \( 2 \), \( R_2(G) \leq \frac{2n}{3} \), while \( r_2(G) \leq \frac{n}{2} \) provided \( G \) is also connected and not \( K_3 \). We also provide bounds for trees \( T \) of order \( n \). We observe that there are, for each \( k \), trees for which \( r_k(T) \leq \frac{n}{2} \), but that the minimum \( R_k(T) \) appears to grow with \( k \); a novel computer algorithm is used to show that \( R_3(T) > \frac{n}{2} \). As expected, these parameters are NP-hard to compute, and we provide a linear-time algorithm for trees for fixed \( k \).
Suppose \( 2n \) voters vote sequentially for one of two candidates. For how many such sequences does one candidate have strictly more votes than the other at each stage of the voting? The answer is \( \binom{2n}{n} \) and, while easy enough to prove using generating functions, for example, only three combinatorial proofs exist, due to Kleitman, Gessel, and Callan. In this paper, we present two new bijective proofs.
In [10], Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs which extends only partially the well-known inequality chain \( \gamma(G) \leq i(G) \leq \beta(G) \leq \Gamma(G) \) between the usual parameters of domination and independence. If a \( k \)-independent set is defined as a subset of vertices inducing in \( G \) a subgraph of maximum degree less than \( k \), we introduce the property which makes a \( k \)-independent set maximal. This leads us to the notion of a \( k \)-star-forming set. The corresponding parameters \( sf_k(G) \) and \( \text{SF}_k(G) \) satisfy \( sf_k(G) \leq i_k(G) \leq \beta_k(G) \leq \text{SF}_k(G) \) where \( i_k(G) \) and \( \beta_k(G) \) are respectively the minimum and the maximum cardinality of a maximal \( k \)-independent set. We initiate the study of \( sf_k(G) \) and \( \text{SF}_k(G) \) and give some results in particular classes of graphs such as trees, chordal graphs, and \( K_{1,r} \)-free graphs.
If \( D \) is a digraph, \( \delta \) its minimum degree, and \( \lambda \) its edge-connectivity, then \( \lambda \leq \delta \). A digraph \( D \) is called super-edge-connected or super-\( \lambda \) if every minimum edge-cut consists of edges adjacent to or from a vertex of minimum degree. Clearly, if \( D \) is super-\( \lambda \), then \( \lambda = \delta \). A digraph without any directed cycle of length \( 2 \) is called an oriented graph. Sufficient conditions for digraphs to be super-edge-connected were given by several authors. However, closely related results for oriented graphs have received little attention until recently. In this paper, we will present some degree sequence conditions for oriented graphs as well as for oriented bipartite graphs to be super-edge-connected.
In this paper we present an efficient exhaustive search strategy on symmetric Boolean functions having the Walsh spectrum values constrained in a range at certain points. Exploiting the structure in Walsh spectrum of a symmetric Boolean function and its relationship with Krawtchouk matrix, we extend the concept of folded vectors and pruning introduced by Gathen and Roche in 1997. The strategy is applied to search for highly nonlinear symmetric Boolean functions and nonlinear symmetric resilient and correlation immune functions. We also experimentally justify that our method provides further efficiency than the search strategy presented by Gathen and Roche.
In 2001, Kristiansen, Hedetniemi, and Hedetniemi [9] first defined the concept of a defensive alliance in a graph, to be a subset \( S \subset V \) of a graph \( G = (V,E) \) having the property that every vertex \( v \in S \) has at most one more neighbor in \( V \setminus S \) than it has in \( S \) (i.e. \( |N[v] \cap S| \geq |N[v] \setminus S| \)). Since then, several other types of alliances have been defined and studied including strong, offensive, global, powerful, and secure alliances. To date, no algorithms or complexity analyses have been developed for alliances in graphs. This is the subject of this paper.
Wythoff quasigroups are a generalization of Wythoff’s game, which in turn is a modification of nim. This paper studies the algebraic structure of Wythoff quasigroups, and in particular the existence of subquasigroups and the question of isomorphism. It is shown that the quasigroups are mutually non-isomorphic, and that there are few possible subquasigroups. The paper concludes with an application to combinatorial games.
A complete enumeration is given of orientable biembeddings involving five of the \( 80 \) Steiner triple systems of order \( 15 \). As a consequence, it follows that each of the \( 80 \) systems has a biembedding in an orientable surface, and precisely \( 78 \) of the systems have orientable self-embeddings.
Given a graph \( G \), the adjacency matrix \( A(G) \), the standard Laplacian \( L(G) \), and the normalized Laplacian \( \mathcal{L}(G) \) have been studied intensively. In this paper, interlacing inequalities are given for each of these three matrices under the two operations of removing an edge or a vertex from \( G \). Examples are given to show that the inequalities are the best possible of their type. In addition, an interlacing result is proven for the adjacency matrix when two vertices of \( G \) are contracted. Among the results given are the following.
Let \( G \) be a graph and let \( H \) be a graph obtained from \( G \) by removing an edge or a vertex of degree \( r \). Let \( \lambda_i \), \( i = 1, 2, \ldots, n \) be the eigenvalues associated with \( A(G) \), \( L(G) \), or \( \mathcal{L}(G) \) and let \( \theta_i \) be the eigenvalues associated with \( A(H) \), \( L(H) \), or \( \mathcal{L}(H) \), where both sets of eigenvalues are in nonincreasing order.
In the case of removing a vertex so that \( H = G – v \), for the normalized Laplacian we have \( \lambda_{i – r + 1} \geq \theta_i \geq \lambda_{i+r} \). For the standard Laplacian we have \( \lambda_i \geq \theta_i \geq \lambda_{i+r} \). In the case of removing an edge so that \( H = G – e \), where \( e \) is an edge incident on a vertex of degree \( 1 \), for the normalized Laplacian we have \( \lambda_i \geq \theta_i \geq \lambda_{i+1} \).
Results are presented on the eternal domination problem: defending a graph from an infinite sequence of attacks, so that each attack is defended by a guard at most distance one from the attack. We first consider the model where at most one guard moves to defend an attack. Our focus is on the relationship between the number of guards and the independence and clique covering numbers of the graph. We establish results concerning which triples of these parameters can be attained by some graph, and determine the exact value of the number of guards for graphs in certain classes. We then turn our attention to the variant of the problem in which every guard can relocate to an adjacent vertex in defence of an attack. We give a linear algorithm to determine the minimum number of guards necessary to defend a tree, and use it to answer another question about defending trees.
A \({generalized\; die}\) is a list \( (x_1,\ldots,x_n) \) of integers. For integers \( n \geq 1, a \leq b \) and \( s \), let \( D(n,a,b,s) \) be the set of all dice with \( a \leq x_1 \leq \ldots \leq x_n \leq b \) and \( \sum x_i = s \). Two dice \( X \) and \( Y \) are \({tied}\) if the number of pairs \( (i,j) \) with \( x_i y_j \). We prove the following: with one exception (unique up to isomorphism), if \( X \neq Y \in D(n,a,b,s) \) are tied dice neither of which ties all other elements of \( D(n,a,b,s) \), then there is a third die \( Z \in D(n,a,b,s) \) which ties neither \( X \) nor \( Y \).
I. Anderson and L. Ellison [7] demonstrated the existence of \( Z \)-cyclic Directed Triplewhist Tournament Designs and \( Z \)-cyclic Ordered Triplewhist Tournament Designs for all primes \( p \equiv 9 \pmod{16} \). It is shown here that their methodology can be generalized completely to deal with primes of the form \( p \equiv (2^k + 1) \pmod{2^{k+1}} \), \( k \geq 4 \).
An \( \mathbb{L}(n,d) \) is a linear space with constant point degree \( n+1 \), lines of size \( n \) and \( n-d \), and with \( v = n^2 – d \) points. Denote by \( b = n^2 + n + z \) the number of lines of an \( \mathbb{L}(n,d) \), then \( z \geq 0 \) and examples are known only if \( z = 0, 1 \) [7]. The linear spaces \( \mathbb{L}(n, d) \) were introduced in [7] in relation with some classification problems of finite linear spaces. In this note, starting from the symmetric configuration \( 45_7 \) of Baker [1], we give an example of \( \mathbb{L}(n,d) \), with \( n=7, d=4 \) and \( z=4 \).
We propose a multilevel cooperative search algorithm to compute upper bounds for \( C_\lambda(v,k,t) \), the minimum number of blocks in a \( t-(v,k,\lambda) \) covering design. Multilevel cooperative search is a search heuristic combining cooperative search and multilevel search. We first introduce a coarsening strategy for the covering design problem which defines reduced forms of an original \( t-(v,k,\lambda) \) problem for each level of the multilevel search. A new tabu search algorithm is introduced to optimize the problem at each level. Cooperation operators between tabu search procedures at different levels include new re-coarsening and interpolation operators. We report the results of tests that have been conducted on \( 158 \) covering design problems. Improved upper bounds have been found for \( 34 \) problems, many of which exhibit a tight gap. The proposed heuristic appears to be a very promising approach to tackle other similar optimization problems in the field of combinatorial design.
A Steiner tree for a set \( S \) of vertices in a connected graph \( G \) is a connected subgraph of \( G \) of smallest size that contains \( S \). The Steiner interval \( I(S) \) of \( S \) is the union of all vertices of \( G \) that belong to some Steiner tree for \( S \). A graph is strongly chordal if it is chordal and has the property that every even cycle of length at least six has an odd chord. We develop an efficient algorithm for finding Steiner intervals of sets of vertices in strongly chordal graphs.
A legal placement of Queens is any placement of Queens on an order \(N\) chessboard in which any two attacking Queens can be separated by a Pawn. The Queens’ independence separation number is the minimum number of Pawns which can be placed on an \(N \times N\) board to result in a separated board on which a maximum of \(m\) independent Queens can be placed. We prove that \(N + k\) Queens can be separated by \(k\) Pawns for large enough \(N\) and provide some results on the number of fundamental solutions to this problem. We also introduce separation relative to other domination-related parameters for Queens, Rooks, and Bishops.
Let \(G\) be a \(2\)-connected graph with maximum degree \(\Delta(G) \geq d\), and let \(x\) and \(z\) be distinct vertices of \(G\). Let \(W\) be a subset of \(V(G) \setminus \{x, z\}\) such that \(|W| \leq d – 1\). Hirohata proved that if \(\max\{d_G(u), d_G(v)\} \geq d\) for every pair of vertices \(u, v \in V(G) \setminus \{x, z\} \cup W\) such that \(d_G(u, v) = 2\), then \(x\) and \(z\) are joined by a path of length at least \(d – |W|\). In this paper, we show that if \(G\) satisfies the conditions of Hirohata’s theorem, then for any given vertex \(y\) such that \(d_G(y) \geq d\), \(x\) and \(z\) are joined by a path of length at least \(d – |W|\) which contains \(y\).
For any positive integer \(k\), there exists a smallest positive integer \(N\), depending on \(k\), such that every \(2\)-coloring of \(1, 2, \ldots, N\) contains a monochromatic solution of the equation \(x + y + kz = 3w\). Based on computer checks, Robertson and Myers in \([5]\) conjectured values for \(N\) depending on the congruence class of \(k\) (mod \(9\)). In this note, we establish the values of \(N\) and find that in some cases they depend on the congruence class of \(k\) (mod \(27\)).
The support of a matrix \(M\) is the \((0, 1)\)-matrix with \(ij\)-th entry equal to \(1\) if the \(ij\)-th entry of \(M\) is non-zero, and equal to \(0\), otherwise. The digraph whose adjacency matrix is the support of \(M\) is said to be the digraph of \(M\). In this paper, we observe some general properties of digraphs of unitary matrices.
The \(k\)-restricted total domination number of a graph \(G\) is the smallest integer \(t_k\), such that given any subset \(U\) of \(k\) vertices of \(G\), there exists a total dominating set of \(G\) of cardinality at most \(t\), containing \(U\). Hence, the \(k\)-restricted total domination number of a graph \(G\) measures how many vertices are necessary to totally dominate a graph if an arbitrary set of \(k\) vertices are specified to be in the set. When \(k = 0\), the \(k\)-restricted total domination number is the total domination number. For \(1 \leq k \leq n\), we show that \(t_k \leq 4(n + k)/7\) for all connected graphs of order \(n\) and minimum degree at least two and we characterize the graphs achieving equality. These results extend earlier results of the author (J. Graph Theory \(35 (2000), 21-45)\). Using these results we show that if \(G\) is a connected graph of order \(n\) with the sum of the degrees of any two adjacent vertices at least four, then \(\gamma_t(G) \leq 4n/7\) unless \(G \in \{C_3, C_5, C_6, C_{10}\}\).
The Szeged index of a graph \(G\) is defined as \(\text{Sz}(G) = \sum_{e=uv \in E(G)} N_u(e|G) N_v(e|G)\), where \(N_u(e|G)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\) and \(N_v(e|G)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). In this article, the Szeged index of some hexagonal systems applicable in nanostructures is computed.
In this paper, we consider the class of impartial combinatorial games for which the set of possible moves strictly decreases. Each game of this class can be considered as a domination game on a certain graph, called the move-graph. We analyze this equivalence for several families of combinatorial games, and introduce an interesting graph operation called iwin and match that preserves the Grundy value. We then study another game on graphs related to the dots and boxes game, and we propose a way to solve it.
Let \(C_n\) denote the cycle with \(n\) vertices, and \(C_n^{(t)}\) denote the graphs consisting of \(t\) copies of \(C_n\), with a vertex in common. Koh et al. conjectured that \(C_n^{(t)}\) is graceful if and only if \(nt \equiv 0,3 \pmod{4}\). The conjecture has been shown true for \(n = 3,5,6,7,9,11,4k\). In this paper, the conjecture is shown to be true for \(n = 13\).
This paper deals with a connection between the universal circuits matrix \([10]\) and the crossing relation \([1,5]\). The value of the universal circuits matrix obtained for \(\overline{\omega}\), where \(\omega\) is an arbitrary feedback function that generates de Bruijn sequences, forms the binary matrix that represents the crossing relation of \(\omega\). This result simplifies the design and study of the feedback functions that generate the de Bruijn sequences and allows us to decipher many inforrnations about the adjacency graphs of another feedback functions. For example, we apply these results to analyze the Hauge-Mykkeltveit classification of a family of de Bruijn sequences \([4]\).
In a given graph \(G\), a set \(S\) of vertices with an assignment of colors is a defining set of the vertex coloring of \(G\), if there exists a unique extension of the colors of \(S\) to a \(\chi(G)\)-coloring of the vertices of \(G\). A defining set with minimum cardinality is called a smallest defining set (of vertex coloring) and its cardinality, the defining number, is denoted by \(d(G, \chi)\). Let \(d(n, r, \chi = k)\) be the smallest defining number of all \(r\)-regular \(k\)-chromatic graphs with \(n\) vertices. Mahmoodian \(et.\; al [7]\) proved that, for a given \(k\) and for all \(n \geq 3k\), if \(r \geq 2(k-1)\) then \(d(n, r, \chi = k) = k-1\). In this paper we show that for a given \(k\) and for all \(n < 3k\) and \(r \geq 2(k – 1)\), \(d(n, r, \chi = k) = k-1\).
A two-colored digraph \(D\) is primitive if there exist nonnegative integers \(h\) and \(k\) with \(h+k > 0\) such that for each pair \((i,j)\) of vertices there exists an \((h, k)\)-walk in \(D\) from \(i\) to \(j\). The exponent of the primitive two-colored digraph \(D\) is the minimum value of \(h + k\) taken over all such \(h\) and \(k\). In this paper, we consider the exponents of families of two-colored digraphs of order \(n\) obtained by coloring the digraph that has the exponent \((n – 1)^2\). We give the tight upper bound on the exponents, and the characterization of the extremal two-colored digraph.
A graceful labeling of a graph \(G\) with \(m\) edges is a function \(f: V(G) \to \{0, \ldots, m\}\) such that distinct vertices receive distinct numbers and \(\{|f(u) – f(v)|: uv \in E(G)\} = \{1, \ldots, m\}\). A graph is graceful if it has a graceful labeling. In \([1]\) this question was posed: “Is there an \(n\)-chromatic graceful graph for \(n \geq 6\)?”. In this paper it is shown that for any natural number \(n\), there exists a graceful graph \(G\) with \(\chi(G) = n\).
A connected graph \(G = (V, E)\) is said to be \((a, d)\)-antimagic, for some positive integers \(a\) and \(d\), if its edges admit a labeling by all the integers in the set \(\{1, 2, \ldots, |E(G)|\}\) such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term \(a\) and the common difference \(d\). Mirka Miller and Martin Ba\'{e}a proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+3}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). The first author of this paper proved that \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 6\). In this paper, we show that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2} , 3)\)-antimagic for \(n \equiv 2 \pmod{4}\), \(n \geq 10\).
Let \(0 \leq p \leq [\frac{r+1}{2}]\) and \(\sigma(K_{r+1}^{-p},n)\) be the smallest even integer such that each \(n\)-term graphic sequence with term sum at least \(\sigma(K_{r+1}^{-p},n)\) has a realization containing \(K_{r+1}^{-p}\) as a subgraph, where \(K_{r+1}^{-p}\) is a graph obtained from a complete graph \(K_{r+1}\) on \(r+1\) vertices by deleting \(p\) edges which form a matching. In this paper, we determine \(\sigma(K_{r+1}^{-p},n)\) for \(r \geq 2, 1 \leq p \leq [\frac{r+1}{2}]\) and \(n \geq 3r + 3\). As a corollary, we also determine \(\sigma(K_{1^*,2^t}n)\) for \(t \geq 1\) and \(n \geq 3s + 6t\), where \(K_{1^*,2^t}\) is an \(r_1\times r_2\times \ldots \times r_{s+t}\) complete \((s + t)\)-partite graph with \(r_1 = r_2 = \ldots = r_s = 1\) and \(r_{s+1} = r_{s+2} = \ldots = r_{s+t} = 2\) and \(\sigma(K_{1^*,2^t},n)\) is the smallest even integer such that each \(n\)-term graphic sequence with term sum at least \(\sigma(K_{1^*,2^t},n)\) has a realization containing \(K_{1^*,2^t}\) as a subgraph.
Let \(n, s_1\) and \(s_2\) be positive integers such that \(1 \leq s_1 \leq n/2, 1 \leq s_2 \leq n/2, s_1 \neq s_2\) and \(gcd(n, s_1, s_2) = 1\). An undirected double-loop network \(G(n;\pm s_1,\pm s_2)\) is a graph \((V, E)\), where \(V = \mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\), and \(E = \{(i \to i+s_1 \mod n), (i\to i-s_1 \mod n), (i\to i+s_2 \mod n), (i\to i-s_2 \mod n) | i = 0, 1, 2, \ldots, n-1\}\). In this paper, a diameter formula is given for an undirected double-loop network \(G(n; \pm s_1, \pm s_2)\). As its application, two new optimal families of undirected double-loop networks are presented.
Anthony J. Macula constructed a \(d\)-disjunct matrix \(\delta(n,d,k)\) in \([1]\), and we now know it is determined by one type of pooling space. In this paper, we give some properties of \(\delta(n,d,k)\) and its complement \(\delta^c(n,d,k)\).
Let \(S\) be a primitive non-powerful signed digraph. The base \(l(S)\) of \(S\) is the smallest positive integer \(l\) such that for all ordered pairs of vertices \(i\) and \(j\) (not necessarily distinct), there exists a pair of \(SSSD\) walks of length \(t\) from \(i\) to \(j\) for each integer \(t \geq l\). In this work, we use \(PNSSD\) to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(n)\) be the largest value of \(l(S)\) for \(S \in\) \(PNSSD\), and \(L(n) = \{l(S) | S \in PNSSD\}\). For \(n \geq 3\), we show \(L(n) = \{2, 3, \ldots, 2n\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop whose bases attain \(l(n)\).
For a graph \(G = (V, E)\) and a binary labeling \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(1)|\). The labeling \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). Any vertex labeling \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^* : E(G) \to \mathbb{Z}_2\) defined by \(f^*(xy) =| f(x) – f(y)|\). Let \(e_f(i) = |f^{*-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by
\[FI(G) = \{|e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G\}.\]
In \([15]\) Lee and Ng conjectured that the friendly index sets of trees will form an arithmetic progression. This conjecture has been mentioned in \([17]\) and other manuscripts. In this paper, we will first determine the friendly index sets of certain caterpillars of diameter four. Then we will disprove the conjecture by presenting an infinite number of trees whose friendly index sets do not form an arithmetic progression.
Let \(S\) be a primitive non-powerful signed digraph of order \(n\). The base of a vertex \(u\), denoted by \(l_S(u)\), is the smallest positive integer \(l\) such that there is a pair of SSSD walks of length \(i\) from \(u\) to each vertex \(v \in V(S)\) for any integer \(t \geq l\). We choose to order the vertices of \(S\) in such a way that \(l_S(1) \leq l_S(2) \leq \ldots \leq l_S(n)\), and call \(l_S(k)\) the \(k\)th local base of \(S\) for \(1 \leq k \leq n\). In this work, we use PNSSD to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(k)\) be the largest value of \(l_S(k)\) for \(S \in\) PNSSD, and \(L(k) = \{l_S(k) | S \in PNSSD\}\). For \(n \geq 3\) and \(1 \leq k \leq n-1\), we show \(I(k) = 2n – 1\) and \(L(k) = \{2, 3, \ldots, 2n-1\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs whose \(k\)th local bases attain \(I(k)\).
Let \(\mathcal{U}_n(k)\) denote the set of all unicyclic graphs on \(n\) vertices with \(k\) (\(k \geq 1\)) pendant vertices. Let \(\diamondsuit_4^k\) be the graph on \(n\) vertices obtained from \(C_4\) by attaching \(k\) paths of almost equal lengths at the same vertex. In this paper, we prove that \(\diamondsuit_4^k\) is the unique graph with the largest Laplacian spectral radius among all the graphs in \(\mathcal{U}_n(k)\), when \(n \geq k + 4\).
For graphs \(G_1, G_2, \ldots, G_m\), the Ramsey number \(R(G_1, G_2, \ldots, G_m)\) is defined to be the smallest integer \(n\) such that any \(m\)-coloring of the edges of the complete graph \(K_n\) must include a monochromatic \(G_i\) in color \(i\), for some \(i\). In this note, we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral \(C_4\), including:\(R(C_4, K_9) \leq 32,
19 \leq R(C_4, C_4, K_4)\leq 22, 31 \leq R(C_4, C_4, C_4, K_4) \leq 50, 52 \leq R(C_4, K_4, K_4) \leq 72, 42 \leq R(C_4, C_4, K_3, K_5) \leq 76, 87\leq R(C_4, C_4, K_4, K_4) \leq 179.\)
We consider the problem of determining the \(Q\)-integral graphs, i.e., the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all \((2, s)\)-semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results concerning \((3, 4)\) and \((3, 5)\)-semiregular bipartite graphs with the same property.
Let \(G\) be a connected graph. For any two vertices \(u\) and \(v\), let \(d(u, v)\) denote the distance between \(u\) and \(v\) in \(G\). The maximum distance between any pair of vertices is called the diameter of \(G\) and denoted by \(diam(G)\). A radio-labeling (or multi-level distance labeling) with span \(k\) for \(G\) is a function \(f\) that assigns to each vertex a label from the set \(\{0, 1, 2, \ldots, k\}\) such that the following holds for any vertices \(u\) and \(v\): \(|f(u) – f(v)| \geq diam(G) – d(u, v) + 1\). The radio number of \(G\) is the minimum span over all radio-labelings of \(G\). The square of \(G\) is a graph constructed from \(G\) by adding edges between vertices of distance two apart in \(G\). In this article, we completely determine the radio number for the square of any path.
Let \(G\) be a simple connected graph containing a perfect matching. \(G\) is said to be BM-extendable if every matching \(M\) whose induced subgraph is a bipartite graph extends to a perfect matching of \(G\). In this paper, for recognizing BM-extendable graphs, we present some conditions in terms of vertex degrees, including the degree sum conditions, the minimum degree conditions, and the Fan-type condition. Furthermore, we show that all these conditions are best possible in some sense.
Let \(u\) be an odd vertex of a bipartite graph \(B\) and suppose that \(f : V(B) \to \mathbb{N}\) is a function such that \(f(u) = \left\lceil d_B(u)/2 \right\rceil\) and \(f(v) = \left\lceil d_B(v)/2 \right\rceil + 1\) for \(v \in V(B) \setminus u\), where \(d_B(v)\) is the degree of \(v\) in \(B\). In this paper, we prove that \(B\) is \(f\)-choosable.
An arc-colored digraph \(D\) is called alternating whenever \(\{(u, v), (v, w)\} \subseteq A(D)\) implies that the color assigned to \((u, v)\) is different from the color of \((v, w)\). In arc-colored digraphs, a set of vertices \(N\) is said to be a kernel by alternating paths whenever it is an independent and dominating set by alternating directed paths (there is no alternating directed path between every pair of its vertices and for every vertex not in \(N\), there exists an alternating path from it to some vertex in \(N\)). With this new concept, we generalize the concept of kernel in digraphs. In this paper, we prove the existence of alternating kernels in possibly infinite arc-colored digraphs with some coloration properties. We also state a bilateral relation between the property of every induced subdigraph of an arc-colored digraph \(D\) of having a kernel by alternating paths and the property of every induced subdigraph of the non-colored digraph \(D\) of having a kernel, with this we enounce several sufficient conditions for \(D\) to have an alternating kernel. Previous results on kernels are generalized.
In this paper, we present a new simple linear-time algorithm for determining the number of spanning trees in the class of complement reducible graphs, also known as cographs. For a cograph \(G\) on \(n\) vertices and \(m\) edges, our algorithm computes the number of spanning trees of \(G\) in \(O(n + m)\) time and space, where the complexity of arithmetic operations is measured under the uniform cost criterion. The algorithm takes advantage of the cotree of the input cograph \(G\) and works by contracting it in a bottom-up fashion until it becomes a single node. Then, the number of spanning trees of \(G\) is computed as the product of a collection of values which are associated with the vertices of \(G\) and are updated during the contraction process. The correctness of our algorithm is established through the Kirchhoff matrix tree theorem, and also relies on structural and algorithmic properties of the class of cographs. We also extend our results to a proper superclass of cographs, namely the \(P_4\)-reducible graphs, and show that the problem of finding the number of spanning trees of a \(P_4\)-reducible graph has a linear-time solution.
This paper extends the concept of paving from finite matroids to matroids of arbitrary cardinality. Afterwards, a paving matroid of arbitrary cardinality is characterized in terms of its collection of closed sets, independent sets, and circuits, respectively.
A Hamiltonian walk in a connected graph \(G\) is a closed walk of minimum length which contains every vertex of \(G\). The Hamiltonian number \(h(G)\) of a connected graph \(G\) is the length of a Hamiltonian walk in \(G\). Let \(\mathcal{G}(n)\) be the set of all connected graphs of order \(n\), \(\mathcal{G}(n, \kappa = k)\) be the set of all graphs in \(\mathcal{G}(n)\) having connectivity \(\kappa = k\), and \(h(n,k) = \{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\). We prove in this paper that for any pair of integers \(n\) and \(k\) with \(1 \leq k \leq n – 1\), there exist positive integers \(a := \min(h;n,k)) = \min\{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\) and \(b := \max(h;n,k)) = \max\{h(G) : G \in \mathcal{G}(n, \kappa = k)\}\) such that \((h;n,k) = \{x \in \mathbb{Z} : a \leq x \leq b\}\). The values of \(\min(h;n,k))\) and \(\max(h(n,k))\) are obtained in all situations.
A well-known result on matchings of graphs is that the intersection of all maximal barriers is equal to the “set A” in the Gallai-Edmonds decomposition. In this paper, we give a generalization of this result to the framework of path-matchings introduced by Cunningham and Geelen. Furthermore, we present a sufficient condition for a graph to have a perfect path-matching.
This paper describes some new methods of constructing rectangular designs from balanced incomplete block (BIB) designs and Hadamard matrices. At the end of the paper, a table of rectangular designs in the range of \(r\),\(k \leq 15\) is given.
An \(n \times n\) sign pattern \(A\) is a spectrally arbitrary pattern if for any given real monic polynomial \(f(x)\) of degree \(n\), there is a real matrix \(B \in Q(A)\) having characteristic polynomial \(f(x)\). In this paper, we give two new classes of \(n \times n\) spectrally arbitrary sign patterns which are generalizations of the pattern \(W_{n}(k)\) defined in [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM Journal on Matrix Analysis and Applications, \(26(2004), 257-271]\).
We show that the power subgroups \(M^{6k}\) (\(k > 1\)) of the Modular group \(M = \text{PSL}(2, \mathbb{Z})\) are subgroups of the groups \(M'(6k, 6k)\). Here, the groups \(M'(6k, 6k)\) (\(k > 1\)) are subgroups of the commutator subgroup \(M’\) of \(M\) of index \(36k^2\) in \(M’\).
Let \(G\) be a simple graph. The double vertex graph \(U_2(G)\) of \(G\) is the graph whose vertex set consists of all \(2\)-subsets of \(V(G)\) such that two distinct vertices \(\{x,y\}\) and \(\{u,v\}\) are adjacent if and only if \(|\{x,y\} \cap \{u,v\}| = 1\) and if \(x = u\), then \(y\) and \(v\) are adjacent in \(G\). In this paper, we consider the exponents and primitivity relationships between a simple graph and its double vertex graph. A sharp upper bound on exponents of double vertex graphs of primitive simple graphs and the characterization of extremal graphs are obtained.
Let \(S_n\) be the set of permutations on \(\{1, \ldots, n\}\) and \(\pi \in S_n\). Let \(d(\pi)\) be the arithmetic average of \(\{|i – \pi(i)| : 1 \leq i \leq n\}\). Then \(d(\pi)/n \in [0, 1/2]\), the expected value of \(d(\pi)/n\) approaches \(1/3\) as \(n\) approaches infinity, and \(d(\pi)/n\) is close to \(1/3\) for most permutations. We describe all permutations \(\pi\) with maximal \(d(\pi)\).
Let \(s^+(\pi)\) and \(s^*(\pi)\) be the arithmetic and geometric averages of \(\{|\pi(i) – \pi(i + 1)| : 1 \leq i 1\). We describe all permutations \(\pi\),\(\sigma\) with maximal \(s^+(\pi)\) and \(s^*(\sigma)\).
A connected graph \(G = (V,E)\) is said to be \((a,d)\)-antimagic, for some positive integers \(a\) and \(d\), if its edges admit a labeling by all the integers in the set \(\{1, 2, \ldots, |E(G)|\}\) such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term \(a\) and the common difference \(d\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n,2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\), and conjectured that \(P(n, k)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for odd \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n,2)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for \(n \equiv 3 \pmod{4}\), \(n \geq 7\).
Sierpiński graphs \(S(n,k)\), \(n, k \in \mathbb{N}\), can be interpreted as graphs of a variant of the Tower of Hanoi with \(k \geq 3\) pegs and \(n \geq 1\) discs. In particular, it has been proved that for \(k = 3\) the graphs \(S(n, 3)\) are isomorphic to the Hanoi graphs \(H_3^n\). In this paper, we will determine the chromatic number, the diameter, the eccentricity of a vertex, the radius, and the centre of \(S(n,k)\). Moreover, we will derive an important invariant and a number-theoretical characterization of \(S(n,k)\). By means of these results, we will determine the complexity of Problem \(1\), that is, the complexity of getting from an arbitrary vertex \(v \in S(n,k)\) to the nearest and to the most distant extreme vertex. For the Hanoi graphs \(H_3^n\), some of these results are new.
In this paper, we will prove that there exist no \([n,k,d]_q\) codes of \(sq^{k-1}-(s+t)q^{k-2}-q^{k-4} \leq d \leq sq^{k-1}-(s+t)q^{k-2}\) attaining the Griesmer bound with \(k \geq 4, 1 \leq s \leq k-2, t \geq 1\), and \(s+t \leq (q+1)\backslash 2\). Furthermore, we will prove that there exist no \([n,k,d]_q\) codes for \(sq^{k-1}-(s+t)q^{k-2}-q^{k-3} \leq d \leq s\) attaining the Griesmer bound with \(k \geq 3\), \(1 \leq s \leq k-2\), \(t \geq 1\), and \(s+t \leq \sqrt{q}-1\). The results generalize the nonexistence theorems of Tatsuya Maruta (see \([7]\)) and Andreas Klein (see \([4]\)) to a larger class of codes.
In this paper, we describe two algorithms to identify the repeating subwords in a given partial word \( w_o = w_0[1,…,n] \). The first algorithm uses the suffix tree and the second algorithm uses the valency tree. Both algorithms take linear time to identify the repeating subwords of a partial word.
We present a class of Coded Petri net languages and study some algebraic properties. The purpose of introduction of this language is to bring out its usefulness in learning theory. We introduce an algorithm for learning a finite coded Petri net language and its running time is bounded by a polynomial function of given inputs.
In this present investigation, the authors obtain Fekete-Szegő’s inequality for certain normalized analytic functions \( f(z) \) defined on the open unit disk. As a special case of this result, Fekete-Szegő’s inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to give a generalization of the Fekete-Szegő inequalities obtained by Srivastava and Mishra and Ma and Minda.
A well-designed interconnection network makes efficient use of scarce communication resources and is used in systems ranging from large supercomputers to small embedded systems on a chip. This paper deals with certain measures of vulnerability in interconnection networks. Let \( G \) be a non-complete connected graph and for \( S \subseteq V(G) \), let \( \omega(G – S) \) and \( m(G – S) \) denote the number of components and the order of the largest component in \( G – S \), respectively. The vertex-integrity of \( G \) is defined as
\[I(G) = \text{min}\{|S| + m(G – S) : S \subseteq V(G)\}.\]
A set \( S \) is called an \( I \)-set of \( G \) if \( I(G) = |S| + m(G – S) \). The rupture degree of \( G \) is defined by
\[r(G) = \text{max}\{\omega(G – S) – |S| – m(G – S) : S \subseteq V(G), \omega(G – S) \geq 2\}.\]
A set \( S \) is called an \( R \)-set of \( G \) if \( r(G) = \omega(G – S) – |S| – m(G – S) \). In this paper, we compute the rupture degree of complete binary trees and a class of meshes. We also study the relationship between an \( I \)-set and an \( R \)-set and find an upper bound for the rupture degree of Hamiltonian graphs.
In this paper, we establish the possibility of embedding a graph as an induced subgraph in an: elegant graph, harmonious graph, felicitous graph, cordial graph, odd-graceful graph, polychrome graph, and strongly c-harmonious graph, each with a given property, leading to prove the NP-completeness of some parameters like: chromatic number, clique number, domination number, and independence number
of these graphs.
This paper describes an approach based on modified invariant moments for recognition of multi-font English characters. The proposed method is independent of size and translation variations and shows better results under noisy conditions. The work treats isolated English characters which are normalized to a size of \( 33 \times 33 \) pixels and the image is thinned. As a pre-classification step, end points and Euler numbers have been estimated from this thinned image of the character. For size and translation invariance, the modified invariant moments suggested by Palaniappan have been evaluated. The system is trained for 7 different font styles with 364 images. A decision tree-based minimum distance nearest neighbor classifier has been adopted for classification. The system is tested for these seven fonts with various sizes of the characters between 8 to 72. A total of 7,280 character images are tested with this system and the success rate is found to be 99.65\%. The method shows encouraging results on multi-font/sized character images.
A Knowledge Based Document Management System (KBDMS) is proposed in this paper to organize, cluster, classify and discover free-text documents. Context sensitive information is discovered by means of word map, sentence map and paragraph map in an intelligent manner in this proposed system. A text learning procedure for the semantic retrieval of text documents is implemented using a hierarchy of self-organizing maps (SOM) and support vector machines (SVM). The hierarchical SOM generates histograms of paragraph maps based on the semantic similarity and these paragraph maps are trained using SVM for classification. The SVM also generates an index for each document given to it. The proposed system is scalable and capable of discovery of documents from a huge amount of free-text documents. It is tested over a maximum of 100,000 text documents with 75-80\% accuracy in the context-sensitive discovery of free-text documents.
The purpose of this paper is to construct the membership functions of performance measures in bulk arrival queuing systems with arrival rate and service rate being fuzzy numbers. Thus, this paper develops the parametric programming approach to derive the membership functions of the steady-state performance measures in bulk arrival queuing systems with varying batch size. On the basis of a cut representation and extension principle, a parametric programming is formulated to describe the family of crisp bulk arrival queues. The performance measures are expressed by membership functions rather than crisp values, which completely conserve the fuzziness of input information when some data of bulk arrival queuing systems are ambiguous.
In order to establish the mathematical basis for connections between molecular structures and physicochemical properties of chemical compounds, some topological indices have been put forward. Among them, the Wiener index is one of the most important topological indices. The sum of distances of all pairs of vertices in a connected graph is known as Wiener index or Wiener number. All structural formulas of chemical compounds are molecular graphs where vertices represent the set of atoms and edges represent chemical bonds. A graph is said to be detour saturated if the addition of any edge results in an increased greatest path length. The characteristic graph of a given benzenoid graph consists of vertices corresponding to hexagonal rings of the graph; two vertices are adjacent if and only if the corresponding rings share an edge. A benzenoid graph is called Cata-condensed if its characteristic graph is a tree. In this paper, we derive Wiener indices for characteristic graphs of benzenoid graphs in the form of hexagonal rings, which are detour-saturated trees.
A vertex \( v \in V(G) \) is said to be a self vertex switching of \( G \) if \( G \) is isomorphic to \( G^v \), where \( G^v \) is the graph obtained from \( G \) by deleting all edges of \( G \) incident to \( v \) and adding all edges incident to \( v \) which are not in \( G \). Two vertices \( u \) and \( v \) in \( G \) are said to be interchange similar if there exists an automorphism \( \alpha \) of \( G \) such that \( \alpha(u) = v \) and \( \alpha(v) = u \). In this paper, we give a characterization for a cut vertex in \( G \) to be a self vertex switching where \( G \) is a connected graph such that any two self vertex switchings, if they exist, are interchange similar.
Pavel Hrnciar and Alfonz Havier \([6]\) introduced a clever idea of transferring labeled pendant edges incident at a vertex of a graceful tree to some other suitable vertex of that tree, so that another graceful tree is obtained. This idea is further explored in this paper to generate graceful lobsters from a graceful caterpillar with \( n \) edges.
In this paper, we study the prime filters of a bounded pseudocomplemented semilattice. We extend some of the results of \([3]\) to pseudocomplemented semilattices. It is observed that the set of all prime filters \( \mathcal{P} \) of a pseudocomplemented semilattice \( S \) is a topology, and it is \( T_0 \) and compact. We also obtain some necessary and sufficient conditions for the subspace of maximal filters to be normal.
A node ranking problem is also called an \({ordered \;coloring\; problem}\) \([6]\), which labels a graph \( G = (V, E) \) with \( C: V \to \{1, 2, \ldots, k\} \) such that for every path between any two nodes \( u \) and \( v \), with \( C(u) = C(v) \), there is a node \( w \) on the path with \( C(w) > C(u) = C(v) \). The value \( C(v) \) is called the \({rank}\) or color of the node \( v \). Node ranking is the problem of finding the minimum \( k \) such that the maximum rank in \( G \) is \( k \). There are two versions of the node ranking problem: off-line and on-line. In the off-line version, all the vertices and edges are given in advance. In the on-line version, the vertices are given one by one in an arbitrary order (say \( v_1, v_2, \ldots, v_n \)) and only the edges of the induced subgraph \( \langle\{v_1, v_2, \ldots, v_i\}\rangle_G \) are known when the rank of \( v_i \) has to be chosen. This paper establishes the node ranking number of complete \( r \)-partite graphs for the off-line version and gives a tight bound for the on-line version with the algorithms to accomplish them in linear time.
Embeddings capabilities play a vital role in evaluating interconnection networks. Wirelength is an important measure of an embedding. As far as the most versatile architecture, the hypercube, is concerned, only approximate estimates of the wirelength of various embeddings are available. This paper presents an optimal embedding of the hypercube into a new architecture called \( k \)-cube necklace, which minimizes wirelength. In addition, this paper gives an exact formula for the minimum wirelength of the hypercube into \( k \)-cube necklace and thereby we solve completely the wirelength problem of the hypercube into \( k \)-cube necklace.
