MacGillivray and Seyffarth (J. Graph Theory \(22 (1996),213-229)\) proved that planar graphs of diameter three have domination number at most ten. Recently it was shown (J. Graph Theory \(40 (2002), 1-25)\) that a planar graph of diameter three and of radius two has domination number at most six while every sufficiently large planer graph of diameter three has domination number at most seven. In this paper we improve on these results. We prove that every planar graph of diameter three and of radius two has total domination number (and therefore domination number) at most five. We show then that every sufficiently large planar graph of diameter three has domination number at most six and this result is sharp, while a planar graph of diameter three has domination number at most nine.
A hypergraph is a generalization of an ordinary graph, in which an edge is not limited to contain exactly two vertices, instead, it can contain an arbitrary number of vertices. A number of desirable properties of database schemes have been shown to be equivalent to hypergraphs. In addition, hypergraph models are very important for cellular mobile communication systems. By applying Pólya’s Enumeration Theorem (PET) twice, the counting series is derived for unlabeled linear acyclic hypergraphs in this paper.
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 and Mendelsohn (1999) proved that for each \(n\geq m\) and each \(r \geq 4\), \(d(n,r, \chi = 3) = 2\). They raised the following question: Is it true that for every \(k\), there exist \(n_0(k)\) and \(r_0(k)\), such that for all \(n \geq n_0(k)\) and \(r \geq r_0(k)\) we have \(d(n,r, \chi = k) = k-1\)? We show that the answer to this question is positive, and we prove 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 subsets of a three-dimensional locally projective planar space which meet every plane either in \(2\) or in \(h, h > 2\), points are studied and classified.
Let \(G_1, G_2\) be simple graphs with \(n_1, n_2\) vertices and \(m_1, m_2\) edges respectively. The Corona graph \(G_1 \circ G_2\) of \(G_1\) with \(G_2\) is obtained by taking one copy of \(G_1\), \(v_1\) copies of \(G_2\) and then joining each vertex of \(G_1\) to all the vertices of a copy of \(G_2\).
For a graph \(G\), by the index of cordiality \(i(G)\) we mean \(\min{|e_f(0)-e_f(1)|}\), where the minimum is taken over all the binary labelings of \(G\) with \(|v_f(0)-v_f(1)|\leq 1\). In this paper, we investigate the cordiality of \(G_1 \circ \overline{K_t}, K_n \circ \overline{K_t}\) and \(G \circ C_t\), where \(G\) is a graph with the index of cordiality \(k\).
In this paper, we give a necessary condition for an odd degree graph to be Skolem-graceful and we prove that if \(G\) is a \((p, q)\) pseudograceful graph such that \(p=q+tl\), then \(G\cup S_m\) is Skolem-graceful for all \(m\geq 1\). Finally, we give some variations on the definition of cordial graphs.
The posets of dimension \(2\) are those posets whose minimal realizations have two elements, that is, which may be obtained as the intersection of two of their linear extensions. Gallai’s decomposition of a poset allows for a simple formula to count the number of the distinct minimal realizations of the posets of dimension \(2\). As an easy consequence, the characterization of M. El-Zahar and of N.W. Sauer of the posets of dimension \(2\), with an unique minimal realization, is obtained.
In this paper we give a new method for constructing modular \(n\)-queens solutions which, in particular, yields nonlinear solutions for all composite \(n\) such that \(\gcd(n,6) = 1\) and all prime \(n \geq 19\).
Path problems in graphs can generally be formulated and solved by using an algebraic structure whose instances are called path algebras. Each type of path problem is characterized by a different instance of the structure. This paper proposes a method for combining already known path algebras into new ones. The obtained composite algebras can be applied to solve relatively complex path problems, such as explicit identification of optimal paths or multi-criteria optimization. The paper presents proofs showing that the proposed construction is correct. Also, prospective applications of composite algebras are illustrated by examples. Finally, the paper explores possibilities of making the construction more general.
Automorphisms of Steiner \(2\)-designs \(S(2,4,37)\) are studied and used to find many new examples. Some of the constructed designs have \(S(2,3,9)\) subdesigns, closing the last gap in the embedding spectrum of \(S(2,3,9)\) designs into \(S(2,4,v)\) designs.
We give a construction for a new family of Group Divisible Designs \((6s + 2, 3, 4; 2, 1)\) using Mutually Orthogonal Latin Squares for all positive integers \(s\). Consequently, we have proved that the necessary conditions are sufficient for the existence of GDD’s of block size four with three groups, \(\lambda_1 = 2\) and \(\lambda_2 = 1\).
For a balanced incomplete block (BIB) design, the following problem is considered: Find \(s\) different incidence matrices of the BIB design such that (i) for \(1 \leq t \leq s-1\), sums of any \(t\) different incidence matrices yield BIB designs and (ii) the sum of all \(s\) different incidence matrices becomes a matrix all of whose elements are one. In this paper, we show general results and present four series of such BIB designs with examples of three other BIB designs.
The extremal matrix problem of symmetric primitive matrices has been completely solved in [Sci. Sinica Ser.A 9(1986) 931-939] and [Linear Algebra Appl.133(1990) 121-131]. In this paper, we determine the maximum exponent in the class of central symmetric primitive matrices, and give a complete characterization of those central symmetric primitive matrices whose exponents actually attain the maximum exponent.
Using a similar framework to \([7]\), we construct a family of relative difference sets in \(P \times ({Z}_{p^2r}^2t)\), where \(P\) is the forbidden subgroup. We only require that \(P\) be an abelian group of order \(p^t\). The construction makes use of character theory and the structure of the Galois ring \(GR(p^{2r}, t)\), and in particular the Teichmüller set for the Galois ring.
For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(h\)-magic if there exists a labeling \(l: E(G) \to \mathbb{Z}_h – \{0\}\) such that the induced vertex set labeling \(l^+: V(G) \to \mathbb{Z}_h\), defined by
\[l^+(v) = \sum\limits_{uv \in E(G)} l(uv)\]
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. In this paper we will identify several classes of zero sum magic graphs and will determine their null sets.
Let \(G\) be a graph, and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(g(x) \leq f(x)\) for all \(x \in V(G)\). A graph \(G\) is called a \((g, f, n)\)-critical graph if \(G-N\) has a \((g, f)\)-factor for each \(N \subseteq V(G)\) with \(|N| = n\). In this paper, a necessary and sufficient condition for a graph to be \((g, f, n)\)-critical is given. Furthermore, the properties of \((g, f, n)\)-critical graphs are studied.
The object of this paper is to give solutions to some of the problems suggested by A.K. Agarwal[\(n\)-color Analogues of Gaussian Polynomials, Ars Combinatoria \(61 (2001), 97-117\)].
For \(n \geq 1\), let \(p(n)\) denote the smallest natural number \(r\) for which the following is true: For \(K\) any finite family of simply connected orthogonal polygons in the plane and points \(x\) and \(y\) in \(\cap\{K : K \in \mathcal{K}\}\), if every \(r\) (not necessarily distinct) members of \(K\) contain a common staircase \(n\)-path from \(x\) to \(y\), then \(\cap\{K : K \in \mathcal{K}\}\) contains such a staircase path. It is proved that \(p(1) = 1, p(2) = 2, p(3) = 4, p(4) = 6\), and \(p(n) \leq 4 + 2p(n – 2)\) for \(n \geq 5\).
The numbers \(p(n)\) are used to establish the following result. For \(\mathcal{K}\) any finite family of simply connected orthogonal polygons in the plane, if every \(3p(n + 1)\) (not necessarily distinct) members of \(\mathcal{K}\) have an intersection which is starshaped via staircase \(n\)-paths, then \(\cap\{K : K \in \mathcal{K}\}\) is starshaped via staircase \((n+1)\)-paths. If \(n = 1\), a stronger result holds.
A \((p,q)\) graph \(G\) is called edge-magic if there exists a bijective function \(f: V(G) \cup E(G) \to \{1,2,\ldots,p+q\}\) such that \(f(u) + f(v) + f(uv) = k\) is a constant for any edge \(uv \in E(G)\). Moreover, \(G\) is said to be super edge-magic if \(f(V(G)) = \{1,2,\ldots, p\}\). The question studied in this paper is for which graphs it is possible to add a finite number of isolated vertices so that the resulting graph is super edge-magic. If it is possible for a given graph \(G\), then we say that the minimum such number of isolated vertices is the super edge-magic deficiency, \(\mu_s(G)\) of \(G\); otherwise we define it to be \(+\infty\).
In this article, we discuss the Helly property and the strong Helly property in hypergraphs. We give a characterization of neighborhood hypergraphs having the Helly and the strong Helly property. These properties are studied in both Cartesian and strong products of hypergraphs.
There are several well-known and important Hamiltonian results for claw-free graphs, but only a few are concerned with quasi-claw-free graphs. In this note, we provide a new sufficient condition for quasi-claw-free Hamiltonian graphs.
A \(d\)-antimagic labeling of a plane graph \(G = (V, E, F)\) is a one-to-one mapping taking the vertices, edges, and faces onto the integers \(1, 2, \ldots, |V(G)| + |E(G)| + |F(G)|\) so that the \(s\)-sided face weights form an arithmetic progression of difference \(d\). This paper describes \(d\)-antimagie labelings for Möbius grids.
There are networks that can be modeled by simple graphs, where edges are perfectly reliable but nodes are subject to failure, e.g. hardwired computer systems. One measure of the “vulnerability” of the network is the connectivity \(\kappa\) of the graph. Another, somewhat related, vulnerability parameter is the component order connectivity \(\kappa_c^{(k)}\), i.e. the smallest number of nodes that must fail in order to ensure that all remaining components have order less than some value \(k\). In this paper we present necessary and sufficient conditions on a 4-tuple \((n,k,a,b)\) for a graph \(G\) to exist having \(n\) nodes, \(\kappa = a\), and \(\kappa_c^{(k)} = b\). Sufficiency of the conditions follows from a specific construction described in our work. Using this construction we obtain ranges of values for the number of edges in a graph having \(n\) nodes, \(\kappa = a\), and \(\kappa_c^{(k)} = b\) thereby obtaining sufficient conditions on the \(5\)-tuple \((n,e,k,a,b)\) for a graph to exist having \(n\) nodes, \(e\) edges, \(\kappa = a\), and \(\kappa_c^{(k)} = b\). In a limited number of special cases, we show the conditions on \((n,e,k,a,b)\) to be necessary as well.
Fishburn, Tanenbaum, and Trenk define the linear discrepancy \(\text{Id}(P)\) of a poset \( P = (V, <_P) \) as the minimum integer \( k \geq 0 \) for which there exists a bijection \( f : V \rightarrow \{1,2,\ldots,|V|\} \) such that \( u <_P v \) implies \( f(u) < f(v) \) and \( u ||_P v \) implies \( |f(u) – f(v)| \leq k \). In their work, they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph. Here we provide partial solutions to some problems formulated in their study about the linear discrepancy and the bandwidth of cocomparability graphs.
To date, investigations on critical sets for a set of mutually orthogonal Latin squares (MOLS) have been carried out only for small orders \( \leq 9 \). In this paper, we deal with a pair of cyclic orthogonal Latin squares of order \( n \), \( n \geq 11 \), \( n \) odd. Through the construction of a uniquely completable set, we give an upper bound on the size of the minimal critical set. In particular, for \( n = 15 \), a critical set achieving this bound is obtained.
We improve the lower bound for the \(\alpha\)-size of trees with maximum degree three.
A graph \( G \) is called \((a,d)\)-\(\text{edge antimagic total}\) \((a, d)\)-\(\text{EAT}\) if there exist integers \( a > 0, d \geq 0 \) and a bijection \( \lambda: V \cup E \rightarrow \{1,2,\ldots,|V| + |E|\} \) such that \(W = \{w(xy) : xy \in E\} = \{a,a+d,\ldots,a+(|E|-1)d\},\) where \( w(xy) = \lambda(x) + \lambda(y) + \lambda(xy) \). An \((a, d)\)-EAT labeling \( \lambda \) of graph \( G \) is \text{super} if \( \lambda(V) = \{1,2,\ldots,|V|\} \). In this paper, we describe how to construct a super \((a, d)\)-EAT labeling on some classes of disconnected graphs, namely \( P_n \cup P_{n+1} \), \( nP_2 \cup P_n \), and \( nP_2 \cup P_{n+2} \), for positive integer \( n \).
A graph \( G(V, E) \) is called a sum graph if there is an injective labeling called sum labeling \( L \) from \( V \) to a set of distinct positive integers \( S \) such that \( xy \in E \) if and only if there is a vertex \( w \in V \) such that \( L(w) = L(x) + L(y) \in S \). In such a case, \( w \) is called a working vertex. Every graph can be made into a sum graph by adding some isolated vertices, if necessary. The smallest number of isolated vertices that need to be added to a graph \( H \) to obtain a sum graph is called the sum number of \( H \); it is denoted by \( \sigma(H) \). A sum labeling which realizes \( H \cup \overline{K_\sigma(G)} \) as a sum graph is called an optimal sum labeling of \( H \).
Sum graph labeling offers a new method for defining graphs and for storing them digitally. Traditionally, a graph is defined as a set of vertices and a set of edges, specified by pairs of vertices which are the endpoints of an edge. To record a graph on a computer, the edges are usually stored either in the form of an adjacency matrix or as a linked list. Using sum graph labeling, we only need to store the set of vertices, together with some additional isolates, if needed. While previously the edges in a graph were specified explicitly, using sum graphs, edges can be specified implicitly.
A sum labeling \( L \) is called an exclusive sum labeling with respect to a subgraph \( H \) of \( G \) if \( L \) is a sum labeling of \( G \) where \( H \) contains no working vertex. The exclusive sum number \( \epsilon(H) \) of a graph \( H \) is the smallest number \( r \) such that there exists an exclusive sum labeling \( L \) which realizes \( H \cup \overline{K_{r}} \) as a sum graph. A labeling \( L \) is an optimal exclusive sum labeling of a graph \( H \) if \( L \) is a sum labeling of \( H \cup K_{\epsilon(H)} \) and \( H \) contains no working vertex. While the exclusive sum number is never smaller than the corresponding sum number of a graph, labeling graphs exclusively has other desirable features which give greater scope for combining two or more labeled graphs.
In this paper, we introduce exclusive sum graph labeling and we construct optimal exclusive sum graph labeling for complete bipartite graphs, paths, and cycles. The paper concludes with a summary of known results in exclusive sum labeling and exclusive sum numbers for several classes of graphs.
A total vertex irregular labeling of a graph \( G \) with \( v \) vertices and \( e \) edges is an assignment of integer labels to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of \( G \), denoted by \( tvs(G) \), is the minimum value of the largest label over all such irregular assignments. In this paper, we consider the total vertex irregular labeling of complete bipartite graphs \( K_{m,n} \) and prove that
\[
tvs(K_{m,n}) \geq \max \left\{ \left\lceil \frac{m+n}{m+1} \right\rceil, \left\lceil \frac{2m+n-1}{n} \right\rceil \right\} \quad \text{if } (m,n) \neq (2,2).
\]
For given graphs \( G \) and \( H \), the Ramsey number \( R(G, H) \) is the smallest natural number \( n \) such that for every graph \( F \) of order \( n \): either \( F \) contains \( G \) or the complement of \( F \) contains \( H \).
This paper investigates the Ramsey number \( R(S_n, W_m) \) of stars versus wheels. We show that if \( m \) is odd, \( n \geq 3 \), and \( m \leq 2n-1 \), then \( R(S_n, W_m) = 3n-2 \). Furthermore, if \( n \) is odd and \( n \geq 5 \), then \( R(S_n, W_m) = 3n – \mu \), where \( \mu = 4 \) if \( m = 2n – 4 \) and \( \mu = 6 \) if \( m = 2n – 8 \) or \( m = 2n – 6 \).
The notions of sum labeling and sum graph were introduced by Harary in 1990. In a sum labeling, a vertex is called a working vertex if its label is equal to the sum of the labels of a pair of two distinct vertices.
A sum labeling of a graph \( G \) is said to be \(\text{exclusive}\) if it is a sum labeling of \( G \) such that \( G \) contains no working vertex. Any connected graph \( G \) will require some additional isolated vertices in order to be labeled exclusively. The smallest number of such isolates is called the exclusive sum number of \( G \); it is denoted by \( \epsilon(G) \). The number of isolates cannot be less than the maximum number of neighbors of any vertex in the graph, that is, at least equal to \( \Delta(G) \), the maximum vertex degree in \( G \). If \( \epsilon(G) = \Delta(G) \), then \( G \) is said to be a \( \Delta \)-\(\text{optimum summable graph}\). An exclusive sum labeling of \( G \) using \( \Delta(G) \) isolates is called a \( \Delta \)-optimum exclusive sum labeling of \( G \).
In this paper, we show that some families of trees are \( \Delta \)-optimum summable graphs. However, this is not true for all trees, and we present an example of a tree that is not a \( \Delta \)-optimum summable graph, giving rise to an open problem.
For graphs \( G_1, G_2, \ldots, G_k \), the (generalized) \(\text{size multipartite Ramsey number}\) \( m_j(G_1, G_2, \ldots, G_k) \) is the least natural number \( m \) such that any coloring of the edges of \( K_{j \times m} \) with \( k \) colors will yield a copy of \( G_i \) in the \( i \)th color for some \( i \). In this note, we determine the exact value of the size multipartite Ramsey number \( m_j(P_s, P_t) \) for \( s = 2, 3 \) and all integers \( t \geq 2 \), where \( P_t \) denotes a path on \( t \) vertices.
Let \( G = (V, E) \) be a graph with \( v \) vertices and \( e \) edges. A \((a, d)\)-\(\text{vertex-antimagic total labeling}\) is a bijection \( \alpha \) from \( V(G) \cup E(G) \) 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 a \(\text{super \((a, d)\)-vertex antimagic total}\). We study basic properties of such labelings and show how to construct such labelings for some families of graphs, such as paths, cycles, and generalized Petersen graphs. We also show that such labeling does not exist for certain families of graphs, such as cycles with at least one tail, trees with an even number of vertices, and all stars.
In this paper, we study the properties of super edge-magic total graphs. In particular, we propose some algorithms to construct new super edge-magic total graphs from the old ones. We also construct a super edge-magic total labeling on certain disconnected graphs, namely \( P_n \cup P_{n+1} \), \( nP_2 \cup P_n \), and \( nP_2 \cup P_{n+2} \).
Let \( G = G(v, e) \) be a finite simple graph with \( v \) vertices and \( e \) edges. An \((a, d)\)-\(\text{edge-antimagic-vertex}\) (EAV) \(\text{labeling}\) is a one-to-one mapping \( f: V(G) \to \{1, 2, \ldots, v\} \) such that for every edge \( xy \in E(G) \), the edge-weight set \(\{f(x) + f(y) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}\) for some positive integers \( a \) and \( d \). An \((a, d)\)-\(\text{edge-antimagic-total labeling}\) is a one-to-one mapping \( f: V(G) \cup E(G) \to \{1, 2, \ldots, v+e\} \) with the property that for every edge \( xy \in E(G) \),\(\{f(x) + f(y) + f(xy) \mid xy \in E(G)\} = \{a, a+d, a+2d, \ldots, a+(e-1)d\}.\) This labeling is called \(\text{super \((a, d)\)-edge-antimagic total labeling}\) if \( f(V(G)) = \{1, 2, \ldots, v\} \). In this paper, we investigate the relationship between the adjacency matrix, \((a, d)\)-edge-antimagic vertex labeling, and super \((a, d)\)-edge-antimagic total labeling, and show how to manipulate this matrix to construct new \((a, d)\)-edge-antimagic vertex labelings and new super \((a, d)\)-edge-antimagic total graphs.
A total labeling of a graph \( G \) with \( p \) vertices and \( q \) edges is a one-to-one mapping from \( V(G) \cup E(G) \) onto \( \{1,2,\ldots,p+q\} \). If the edge-weights (resp. vertex-weights) form an arithmetic progression starting from \( a \) and having common difference \( d \), then the labeling is called an \( (a,d) \)-edge (resp. vertex) – antimagic total labeling. In this paper, we consider such labeling applied to the generalized Petersen graph.
A simple graph \( G = (V,E) \) admits an \( H \)-\({covering}\) if every edge in \( E \) belongs to a subgraph of \( G \) isomorphic to \( H \). In this case, we say that \( G \) is \( H \)-\({magic}\) if there is a total labeling \( f : V \cup E \rightarrow \{1,2,\ldots,|V|+|E|\} \) such that for each subgraph \( H’ = (V’,E’) \) of \( G \) isomorphic to \( H \),\(\sum_{v \in V’} f(v) + \sum_{e \in E’} f(e)\)
is constant. When \( f(V) = \{1,\ldots,|V|\} \), we say that \( G \) is \( H \)-\({supermagic}\).We study \( H \)-magic graphs for several classes of connected graphs. We also provide constructions of infinite families of \( H \)-magic graphs for an arbitrary given graph \( H \).
Let \( \lambda \) be an edge-magic total (EMT) labeling of graph \( G(V, E) \). Let \( W \subset V(G) \cup E(G) \). Any restriction of \( \lambda \) to \( W \) is called a \({partial\; EMT \;labeling}\) on \( G \). A partial EMT labeling \( \pi \) is a critical set in \( \lambda \) if \( \lambda \) is the only edge-magic total labeling having \( \pi \) as its partial EMT labeling, and no proper restriction of \( \pi \) satisfies the first condition. In this paper, we study the property of critical sets in such a labeling. We determine critical sets in an EMT labeling for a given graph \( G \).
A \( (p,q) \) graph \( G \) is called edge-magic if there exists a bijective function \( f : V(G) \cup E(G) \rightarrow \{1,2,\ldots,p+q\} \) such that \( f(u) + f(v) + f(uv) \) is a constant for each edge \( uv \in E(G) \). Also, \( G \) is said to be super edge-magic if \( f(V(G)) = \{1,2,\ldots, p\} \). Furthermore, the super edge-magic deficiency, \( \mu_s(G) \), of a graph \( G \) is defined to be either the smallest nonnegative integer \( n \) with the property that the graph \( G \cup nK_1 \) is super edge-magic or \( +\infty \) if there exists no such integer \( n \).
In this paper, the super edge-magic deficiency of certain forests and 2-regular graphs is computed, which in turn leads to some conjectures on the super edge-magic deficiencies of graphs in these classes. Additionally, some edge-magic deficiency analogues to the super edge-magic deficiency results on forests are presented.
Suppose \( G = (V,E,F) \) is a finite plane graph with vertex set \( V(G) \), edge set \( E(G) \), and face set \( F(G) \). A bijection \( \lambda: V(G) \cup E(G) \cup F(G) \rightarrow \{1,2,3,\ldots,|V(G)| + |E(G)| + |F(G)|\} \) is called a labeling of type \( (1,1,1) \). The weight of a face under a labeling is the sum of the labels (if present) carried by that face and the edges and vertices surrounding it. A labeling of a plane graph \( G \) is called \( d \)-\({antimagic}\) if for every number \( s \geq 3 \), the set of \( s \)-sided face weights is\(W_s = \{a_s + id: 0 \leq i \leq f_s\}\) for some integers \( a_s \) and \( d \) (\( a > 0 \), \( d \geq 0 \)), where \( f_s \) is the number of \( s \)-sided faces. We allow different sets \( W_s \) for different \( s \).
In this paper, we deal with \( d \)-\({antimagic}\) labelings of type \( (1,1,1) \) for a special class of plane graphs \( C_a^b \) and we show that a \( C_a^b \) graph has \( d \)-antimagic labeling for \( d \in \{a-2,a-1,a+1,a+2\} \).
A \( V(m,t) \) leads to \( m \) idempotent pairwise orthogonal Latin squares of order \( (m+1)t+1 \) with one common hole of order \( t \). \( V(m,t) \)’s can also be used to construct perfect Mendelsohn designs and optimal optical orthogonal codes. For \( 3 \leq m \leq 8 \), the spectrum for \( V(m,t) \) has been determined. In this article, we investigate the existence of \( V(m,t) \) with \( m = 9 \) and show that a \( V(9,t) \) always exists in \( GF(q) \) for any prime power \( q = 9t + 1 \) with the exception of \( q = 73 \) and one possible exception of \( q = 5^6 \).
A Vertex Magic Total Labeling of a graph \( G \) is a one-to-one map \( \lambda \) from \( E(G) \cup V(G) \) onto the set of integers \( \{1, 2, \ldots, e + v\} \) such that for all \( x \in V \) we have \(\lambda(x) + \sum \lambda(xy) = h\) for some constant \( h \), where the sum is taken over all vertices \( y \) adjacent to \( x \). In this paper, we present several theorems on the existence of such labelings for multipartite graphs and give constructions for labelings for two infinite families of complete tripartite graphs, namely \( K_{1,n,n} \) for odd \( n \) and \( K_{2,n,n} \) for \( n \equiv 3 \pmod{4} \).
In this paper, we develop a polynomial time algorithm to determine the cyclic edge connectivity of a \(k\)-regular graph for \(k \geq 3\). The time complexity of the algorithm is bounded by \(O(k^{11}|V|^8)\), in particular, it is \(O(|V|^8)\) for cubic graphs.
For each integer \(m \geq 1\), consider the graph \(G_m\) whose vertex set is the set \(\mathbb{N} = \{0,1,2,\ldots\}\) of natural numbers and whose edges are the pairs \(xy\) with \(y = x+m\), \(y = x-m\), \(y = mx\), or \(y = \frac{x}{m}\). Our aim in this note is to show that, for each \(m\), the graph \(G_m\) contains a Hamilton path. This answers a question of Lichiardopol.
Given a partial \(K_4\)-design \((X, {P})\), if \(x \in X\) is a vertex which occurs in exactly one block of \({P}\), then call \(x\) a free vertex. In this paper, a technique is described for obtaining a cubic embedding of any partial \(K_4\)-design with the property that every block in the partial design contains at least two free vertices.
The average distance \(\mu(D)\) of a strong digraph \(D\) is the average of the distances between all ordered pairs of distinct vertices of \(D\). Plesnik \([6]\) proved that if \(D\) is a strong tournament of order \(n\), then \(\mu(D) \leq \frac{n+4}{6} + \frac{1}{n}\). In this paper, we show that if \(D\) is a \(k\)-connected tournament of order \(n\), then \(\mu(D) \leq \frac{n}{6k} + \frac{19}{6} + \frac{k}{n}\). We demonstrate that, apart from an additive constant, this bound is best possible.
A subset \(U\) of a set \(S\) with a binary operation is called avoidable if \(S\) can be partitioned into two subsets \(A\) and \(B\) such that no element of \(U\) can be written as a product of two distinct elements of \(A\) or as the product of two distinct elements of \(B\). The avoidable sets of the bicyclic inverse semigroup are classified.
Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by
\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]
Let \(\alpha, \beta\) be any numbers. Given an initial sequence \(a_{0,m}\) (\(m = 0,1,2,\ldots\)), define the sequences \(a_{n,m}\) (\(n \geq 1\)) recursively by
\[a_{n,m} = \alpha a_{n-1,m} + \beta a_{n-1,m+1}; \quad \text{for n} \geq 1, m \geq 0.\]
We call the matrix \((a_{n,m})_{n,m\geq 0}\) an generalized Seidel matrix with a parameter pair \((\alpha, \beta)\). If \(\alpha = \beta = 1\), then this matrix is the classical Seidel matrix. For various different parameter pairs \((\alpha, \beta)\) we will impose some evenness or oddness conditions on the exponential generating functions of the initial sequence \(a_{0,m}\) and the final sequence \(a_{n,0}\) of a generalized Seidel matrix (i.e., we require that these generating functions or certain related functions are even or odd). These conditions imply that the initial sequences and final sequences are equal to well-known classical sequences such as those of the Euler numbers, the Genocchi numbers, and the Springer numbers.
As applications, we give a straightforward proof of the continued fraction representations of the ordinary generating functions of the sequence of Genocchi numbers. And we also get the continued fractions representations of the ordinary generating functions of the Genocchi polynomials, Bernoulli polynomials, and Euler polynomials. Lastly, we give some applications of congruences for the Euler polynomials.
Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(f: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(\overline{f}(uv) = |f(u) – f(v)|\). Let \(v_f(0), v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0), e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – v_f(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).
In this paper, we give necessary and sufficient conditions for the cordiality of the \(t\)-ply \(P_t(u,v)\), i.e. a thread of ply number \(t\).
A Jacobi polynomial was introduced by Ozeki. It corresponds to the codes over \(\mathbb{F}_2\). Later, Bannai and Ozeki showed how to construct Jacobi forms with various index using a Jacobi polynomial corresponding to the binary codes. It generalizes Broué-Enguehard map. In this paper, we study Jacobi polynomial which corresponds to the codes over \(\mathbb{F}_{2f}\). We show how to construct Jacobi forms with various index over the totally real field. This is one of extension of Broué-Enguehard map.
The paper contains two main results. First, we obtain the chromatic polynomial on the \(n \times m\) section of the square lattice, solving a problem proposed by Read and Tutte \([5]\), the chromatic polynomial of the bracelet square lattice, and we find a recurrent-constructive process for the matrices of the \(k\)-colourings. The key concept for obtaining the inductive method is the compatible matrix.
Our second main result deals with the compatible matrix as the adjacency matrix of a graph. This represents a family of graphs, which is described.
Let \(G = (V, E)\) be a simple graph. For any real valued function \(f: V \to \mathbb{R}\), the weight of \(f\) is defined as \(f(V) = \sum f(v)\), over all vertices \(v \in V\). For positive integer \(k\), a total \(k\)-subdominating function (TkSF) is a function \(f: V \to \{-1,1\}\) such that \(f(N(v)) \geq k\) for at least \(k\) vertices \(v\) of \(G\). The total \(k\)-subdomination number \(\gamma^t_{ks}(G)\) of a graph \(G\) equals the minimum weight of a TKSF on \(G\). In the special case where \(k = |V|\), \(\gamma^t_{ks}(G)\) is the signed total domination number \([5]\). We research total \(k\)-subdomination numbers of some graphs and obtain a few lower bounds of \(\gamma^t_{ks}(G)\).
A convex hull of a set of points \(X\) is the minimal convex set containing \(X\). A box \(B\) is an interval \(B = \{x | x \in [a,b], a,b \in \mathbb{R}^n\}\). A box hull of a set of points \(X\) is defined to be the minimal box containing \(X\). Because both convex hulls and box hulls are closure operations of points, classical results for convex sets can naturally be extended for box hulls. We consider here the extensions of theorems by Carathéodory, Helly, and Radon to box hulls and obtain exact results.
The point-distinguishing chromatic index of a graph \(G = (V, E)\) is the smallest number of colors assigned to \(E\) so that no two different points are incident with the same color set. In this paper, we discuss the bounds of the point-distinguishing chromatic indices of graphs resulting from the graph operations. We emphasize that almost all of these bounds are best possible.
If \(G\) is a bipartite graph with bipartition \((X,Y)\), a subset \(S\) of \(X\) is called a one-sided dominating set if every vertex \(y \in Y\) is adjacent to some \(x \in S\). If \(S\) is minimal as a one-sided dominating set (i.e., if it has no proper subset which is also a one-sided dominating set), it is called a bipartite dominating set (see \([4], [5]\), and \([6]\)). We study bipartite dominating sets in hypercubes.
Let \(u,v\) be distinct vertices of a multigraph \(G\) with degrees \(d_u\) and \(d_v\), respectively. The number of edge-disjoint \(u,v\)-paths in \(G\) is bounded above by \(\min\{d_u,d_v\}\). A multigraph \(G\) is optimally edge-connected if for all pairs of distinct vertices \(u\) and \(v\) this upper bound is achieved. If \(G\) is a multigraph with degree sequence \(D\), then we say \(G\) is a realisation of \(D\). We characterise degree sequences of multigraphs that have an optimally edge-connected realisation as well as those for which every realisation is optimally edge-connected.
The \(associated \;graph \;of\; a\) \((0,1)\)-\(matrix\) has as its vertex set the lines of the matrix with vertices adjacent whenever their lines intersect at \(a\) \(1\). This association relates the \((0,1)\)-matrix and bipartite graph versions of the König-Egervary Theorem. We extend this graph association to higher dimensional matrices. We characterize these graphs, modulo isolated vertices, using a coloring in which every path between each pair of vertices contains the same two colors. We rely on previous results about \(p\)-dimensional gridline graphs, where vertices are \(1\)’s in a higher dimensional matrix and vertices are adjacent whenever they are on a common line. Also important is the dual property that the doubly iterated clique graph of a diamond- and simplicial vertex-free graph is isomorphic to the original.
Since Cohen introduced the notion of competition graph in \(1968\), various variations have been defined and studied by many authors. Using the combinatorial properties of the adjacency matrices of digraphs, Cho \(et\; al\). \([2]\) introduced the notion of a \(m\)-step competition graph as a generalization of the notion of a competition graph. Then they \([3]\) computed the \(2\)-step competition numbers of complete graphs, cycles, and paths. However, it seems difficult to compute the \(2\)-step competition numbers even for the trees whose competition numbers can easily be computed. Cho \(et\; al\). \([1]\) gave a sufficient condition for a tree to have the \(2\)-step competition number two. In this paper, we show that this sufficient condition is also a necessary condition for a tree to have the \(2\)-step competition number two, which completely characterizes the trees whose \(2\)-step competition numbers are two. In fact, this result turns out to characterize the connected triangle-free graphs whose \(2\)-step competition numbers are two.
In this paper, we are interested in lexicographic codes which are greedily constructed codes. For an arbitrary length \(n\), we shall find the basis of quaternary lexicographic codes, for short, lexicodes, with minimum distance \(d_m = 4\). Also, using a linear nim sum of some bases (such a vector is called the testing vector), its decoding algorithm will be found.
A maximal-clique partition of a graph is a family of its maximal complete sub-graphs that partitions its edge set. Many graphs do not have a maximal-clique partition, while some graphs have more than one. It is harder to find graphs in which maximal-clique partitions have different sizes. \(L(K_5)\) is a well-known example. In \(1982\), Pullman, Shank, and Wallis \([9]\) asked if there is a graph with fewer vertices than \(L(K_5)\) with this property. This paper confirms that there is no such graph.
Can an arbitrary graph be embedded in Euclidean space so that the isometry group of its vertex set is precisely its graph automorphism group? This paper gives an affirmative answer, explores the number of dimensions necessary, and classifies the outerplanar graphs that have such an embedding in the plane.
In this note, we consider arithmetic properties of the function
\[K(n)=\frac{(2n)!(2n+2)!}{(n-1)!(n+1)!^2(n+2)!}\]
which counts the number of two-legged knot diagrams with one self-intersection and \(n-1\) tangencies. This function recently arose in a paper by Jacobsen and Zinn-Justin on the enumeration of knots via a transfer matrix approach. Using elementary number theoretic techniques, we prove various results concerning \(K(n)\), including the following:
A Halin graph is a plane graph \(H = T \cup C\), where \(T\) is a tree with no vertex of degree two and at least one vertex of degree three or more, and \(C\) is a cycle connecting the pendant vertices of \(T\) in the cyclic order determined by the drawing of \(T\). In this paper we determine the list chromatic number, the list chromatic index, and the list total chromatic number (except when \(\Delta = 3\)) of all Halin graphs, where \(\Delta\) denotes the maximum degree of \(H\).
In \([4]\) Fan Chung Graham investigates the notion of graph labelings and related bandwidth and cutwidth of such labelings when the host graph is a path graph. Motivated by problems presented in \([4]\) and our investigation of designing efficient virtual path layouts for communication networks, we investigate in this note labeling methods on graphs where the host graph is not restricted to a particular kind of graph. In \([2]\) authors introduced a metric on the set of connected simple graphs of a given order which represents load on edges of host graph under some restrictions on bandwidth of such labelings. In communication networks this translates into finding mappings between guest graph and host graph in a way that minimizes the congestion while restricting the delay. In this note, we present optimal mappings between special \(n\)-vertex graphs in \(\mathcal{G}_n\), and compute their distances with respect to the metric introduced in \([2]\). Some open questions are also presented.
We discuss several equivalent definitions of matroids, motivated by the single forbidden minor of matroid basis clutters.
Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type \((1,2)\) or \((2,1)\). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers.
In the present paper we give a complete solution for the number of permutations avoiding a pair of patterns of type \((1,2)\) or \((2,1)\). We also conjecture the number of permutations avoiding the patterns in any set of three or more such patterns.
Let \(k \geq 1\) be an integer and let \(G\) be a graph of order \(p\). A set \(S\) of vertices in a graph is a total \(k\)-dominating set if every vertex of \(G\) is within distance at most \(k\) from some vertex of \(S\) other than itself. The smallest cardinality of such a set of vertices is called the total \(k\)-domination number of the graph and is denoted by \(\gamma_k^t(G)\). It is well known that \(\gamma_k^t(G) \leq \frac{2p}{2k+1}\) for \(p \leq 2k + 1\). In this paper, we present a characterization of connected graphs that achieve the upper bound. Furthermore, we characterize the connected graph \(G\) with \(\gamma_k^t(G) + \gamma_k^t(\overline{G}) = \frac{2p}{2k+1} + 2\).
A rational number \(\frac{p}{q}\) is said to be a closest approximation to a given real number \(\alpha\) provided it is closer to \(\alpha\) than any other rational number with denominator at most \(q\). We determine the sequence of closest approximations to \(\alpha\), giving our answer in terms of the simple continued fraction expansion of \(\alpha\).
We describe an algorithm that uses \( O(n) \) arithmetic operations for computing the determinant of the matrix \( M = (A + \alpha I) \), where \( A \) is the adjacency matrix of an order \( n \) tree. Combining this algorithm with interpolation, we derive a simple algorithm requiring \( O(n^2) \) arithmetic operations to find the characteristic polynomial of the adjacency matrix of any tree. We apply our algorithm and recompute a 22-degree characteristic polynomial, which had been incorrectly reported in the quantum chemistry literature.
A vertex set \( D \) of a graph \( G \) is a dominating set if every vertex not in \( D \) is adjacent to some vertex in \( D \). The domination number \( \gamma \) of a graph \( G \) is the minimum cardinality of a dominating set in \( G \). In 1989, Brigham and Dutton [1] proved
\[
\gamma(G) \leq \left\lceil\frac{3n-g}{6}\right\rceil
\]
for each graph \( G \) of order \( n \), minimum degree \( \delta \geq 2 \), and girth \( g \geq 5 \). For this class of graphs, Volkmann [8] recently gave the better bound
\[
\gamma(G) \leq \left\lceil\frac{3n-g-6}{8}\right\rceil
\]
if \( G \) is neither a cycle nor one of two exceptional graphs. If \( G \) is a graph of order \( n \), minimum degree \( \delta \geq 2 \), girth \( g \geq 5 \), then we show in this paper that
\[
\gamma(G) \leq \left\lceil\frac{3n-g-9}{6}\right\rceil
\]
if \( G \) is neither a cycle nor one of 40 exceptional graphs of order between 8 and 21.
A caterpillar \( R \) is a tree with the property that after deleting all vertices of degree 1, we obtain a path \( P \) or a single vertex. The path \( P \) is called the spine of caterpillar \( R \). If the spine has length 3 and \( R \) contains vertices of degrees \( r, s, 2, 2 \), where \( r, s > 2 \), then we say that \( R \) is a \( \{r, s, 2, 2\} \)-caterpillar of diameter 5. We completely characterize \( \{r, s, 2, 2\} \)-caterpillars of diameter 5 on \( 4k + 2 \) vertices that factorize \( K_{4k+2} \).
In the theory of cocyclic self-dual codes, three types of equivalences are encountered: cohomology or the equivalence of cocycles, Hadamard equivalence or the equivalence of Hadamard matrices, and the equivalence of binary linear codes. There are some results relating the latter two equivalences, see Ozeki [12], but not when the Hadamard matrices are un-normalised.
Recently, Horadam [9] discovered shift action, whereby every finite group \( G \) acts as a group of automorphisms of \( Z = Z^2(G, C) \), the finite abelian group of cocycles from \( G \times G \to C \), for each abelian group \( C \). These automorphisms fix the subgroup of coboundaries \( B \leq Z \) setwise. This shift action of \( G \) on \( Z \) partitions each cohomology class of \( Z \).
Here we show that shift-equivalent cocycles generate equivalent Hadamard matrices and that shift-equivalent cocyclic Hadamard matrices generate equivalent binary linear codes.
New identities involving the Catalan sequence ordinary generating function are developed, and a previously known one established from first principles using a hypergeometric approach.
We examine words \( w \) satisfying the following property: if \( x \) is a subword of \( w \) and \( |x| \) is at least \( k \) for some fixed \( k \), then the reversal of \( x \) is not a subword of \( w \).
For constructing routes in mobile ad-hoc networks (MANET) and sensor networks, it is highly desirable to perform primitive computations locally. If a network can be represented in the doubly connected edge list (DCEL) data structure, then many operations can be done locally. However, the DCEL data structure can be used to represent only planar graphs. In this paper, we propose an extended version of the DCEL data structure called ExtDCEL that can be used for representing non-planar graphs as well as their planar components. The proposed data structure can be used to represent geometric networks in mobile computing that include unit disk graphs, Gabriel graphs, and constrained Delaunay triangulations. We show how the proposed data structure can be used to implement a hybrid greedy face routing algorithm in optimum \( O(m) \) time, where \( m \) is the number of edges in the unit disk graph. We also report on the implementation of several routing algorithms for mobile computing by using the proposed data structure.
In this paper, we study the decomposition of the graph \( (\lambda D_v)^{+\alpha} \) into extended cyclic triples, for all \( \lambda \geq \alpha \). By an extended cyclic triple, we mean a loop, a loop with symmetric arcs attached (known as a lollipop), or a directed \( 3 \)-cycle (known as a cyclic triple).
In this paper, we consider the problem of the non-existence of some orthogonal arrays (O-arrays) of strength four with two levels, the number of constraints \( k \) satisfying \( 4 \leq k \leq 32 \), and index set \( \lambda \) where \( 1 \leq \lambda \leq 64 \).
We give a constructive proof that a planar graph on \( n \) vertices with degree of regularity \( k \) exists for all pairs \( (n,k) \) except for two pairs \( (7,4) \) and \( (14,5) \). We continue this theme by classifying all strongly regular planar graphs, and then consider a new class of graphs called \( 2 \)-\({strongly\; regular}\). We conclude with a conjectural classification of all planar \( 2 \)-strongly regular graphs.
This paper answers the question as to whether every natural number \( n \) is realizable as the number of ones in the top portion of rows of a general binary Pascal triangle. Moreover, the minimum number \( \kappa(n) \) of rows is determined so that \( n \) is realizable.
A \( (p,q) \)-graph \( G \) is said to be \(\textbf{edge-graceful}\) if the edges can be labeled by \( 1,2,\ldots, q \) so that the vertex sums are distinct, mod \( p \). It is shown that if a tree \( T \) is edge-graceful, then its order must be odd. Lee conjectured that all trees of odd orders are edge-graceful. The conjecture is still unsettled. In this paper, we give the state of the progress toward this tantalizing conjecture.
We use a new technique for decomposition of complete graphs with even number of vertices based on \( 2n \)-cyclic blended labeling to show that for every \( k > 1 \) odd, and every \( d \), \( 3 \leq d \leq 2^qk – 1 \), there exists a spanning tree of diameter \( d \) that factorizes \( K_{2^qk} \).
A constant composition code of length \( n \) over a \( k \)-ary alphabet has the property that the numbers of occurrences of the \( k \) symbols within a codeword is the same for each codeword. These specialize to constant weight codes in the binary case, and permutation codes in the case that each symbol occurs exactly once. Constant composition codes arise in powerline communication and balanced scheduling, and are used in the construction of permutation codes. Using exhaustive and probabilistic clique search, and by applying theorems and constructions in past literature, we generate tables which summarize the best known lower bounds on constant composition codes for (i) \( 3 \leq k \leq 8 \), (ii) \( k = 3 \), \( 9 \leq n \leq 12 \), and (iii) various other interesting parameters with \( n \geq 9 \).
In this paper, we develop a computational method for constructing transverse \( t \)-designs. An algorithm is presented that computes the \( G \)-orbits of \( k \)-element subsets transverse to a partition \( \mathcal{H} \), given that an automorphism group \( G \) is provided. We then use this method to investigate transverse Steiner quadruple systems. We also develop recursive constructions for transverse Steiner quadruple systems, and we provide a table of existence results for these designs when the number of points \( v \leq 24 \). Finally, some results on transverse \( t \)-designs with \( t > 3 \) are also presented.
A vertex-magic total labeling of a graph \( G(V, E) \) is defined as a one-to-one mapping from \( V \cup E \) to the set of integers \( \{1,2,\ldots,|V| + |E|\} \) with the property that the sum of the label of a vertex and the labels of all edges incident to this vertex is the same constant for all vertices of the graph. A supermagic labeling of a graph \( G(V, E) \) is defined as a one-to-one mapping from \( E \) to the set of integers \( \{1, 2,\ldots,|E|\} \) with the property that the sum of the labels of all edges incident to a vertex is the same constant for all vertices of the graph.
In this paper, we present a technique for constructing vertex-magic total labelings of products of certain vertex-magic total \( r \)-regular graphs \( G \) and certain \( 2_s \)-regular supermagic graphs \( H \). \( H \) has to be decomposable into two \( s \)-regular factors and if \( r \) is even, \( |H| \) has to be odd.
At each vertex in a Cayley map, the darts emanating from that vertex are labeled by a generating set of a group. This generating set is closed under inverses. Two classes of Cayley maps are balanced and antibalanced maps. For these cases, the distributions of the inverses about the vertex are well understood. For a balanced Cayley map, either all the generators are involutions or each generator is directly opposite across the vertex from its inverse. For an antibalanced Cayley map, there is a line of reflection in the tangent plane of the vertex so that the inverse generator for each dart label is symmetric across that line. An \( e \)-balanced Cayley map is a recent generalization that has received much study, see for example [2, 6, 7, 13]. In this note, we examine the symmetries of the inverse distributions of \( e \)-balanced maps in a manner analogous to those of balanced and antibalanced maps.
In [Kit1] Kitaev discussed simultaneous avoidance of two \(3\)-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such \(n\)-permutations are \(2^{n-1}\), the number of involutions in \(S_n\), and \(2^{E_n}\), where \(E_n\) is the \(n\)-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases.
To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form \(x-y-z\) (a classical \(3\)-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a \(3\)-pattern, begin with a certain pattern, and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized \(3\)-pattern and beginning and ending with increasing or decreasing patterns.
A graph \(G\) is called integral or Laplacian integral if all the eigenvalues of the adjacency matrix \(A(G)\) or the Laplacian matrix \(Lap(G) = D(G) – A(G)\) of \(G\) are integers, where \(D(G)\) denotes the diagonal matrix of the vertex degrees of \(G\). Let \(K_{n,n+1} \equiv K_{n+1,n}\) and \(K_{1,p}[(p-1)K_p]\) denote the \((n+1)\)-regular graph with \(4n+2\) vertices and the \(p\)-regular graph with \(p^2 + 1\) vertices, respectively. In this paper, we shall give the spectra and characteristic polynomials of \(K_{n,n+1} \equiv K_{n+1,n}\) and \(K_{1,p}[(p-1)K_p]\) from the theory on matrices. We derive the characteristic polynomials for their complement graphs, their line graphs, the complement graphs of their line graphs, and the line graphs of their complement graphs. We also obtain the numbers of spanning trees for such graphs. When \(p = n^2 + n + 1\), these graphs are not only integral but also Laplacian integral. The discovery of these integral graphs is a new contribution to the search of integral graphs.
Balakrishnan et al. \([1, 2]\) have shown that every graph is a subgraph of a graceful graph and an elegant graph. Also Liu and Zhang \([4]\) have shown that every graph is a subgraph of a harmonious graph. In this paper we prove a generalization of these two results that any given set of graphs \(G_1,G_1,\ldots,G_i\) can be packed into a graceful/harmonious/elegant graph.
We consider compositions or ordered partitions of the natural number n for which the largest (resp. smallest) summand occurs in the first position of the composition.
Let \(m \geq 4\) be a positive integer and let \({Z}_m\) denote the cyclic group of residues modulo \(m\). For a system \(L\) of inequalities in \(m\) variables, let \(R(L;2)\) (\(R(L;{Z}_m)\)) denote the minimum integer \(N\) such that every function \(\Delta: \{1,2,\ldots,N\} \to \{0,1\}\) (\(A: \{1,2,\ldots,N\} \to {Z}_m\)) admits a solution of \(L\), say \((z_1,\ldots,z_m)\), such that \(\Delta(x_1) = \Delta(x_2) = \cdots = \Delta(x_m)\) (such that \(\sum_{i=1}^{m}\Delta(x_i) = 0\)). Define the system \(L_1(m)\) to consist of the inequality \(x_2 – x_1 \leq x_m – x_3\), and the system \(L_2(m)\) to consist of the inequality \(x_{m – 2}-x_{1} \leq x_m – x_{m-1}\); where \(x_1 < x_2 < \cdots < x_m\) in both \(L_1(m)\) and \(L_2(m)\). The main result of this paper is that \(R(L_1(m);2) = R(L_1(m);{Z}_m) = 2m\), and \(R(L_2(m);2) = 6m – 15\). Furthermore, we support the conjecture that \(R(L_1(m);2) = R(L_1(m);{Z}_m)\) by proving it for \(m = 5\).
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 \(d(G, \chi)\). We study the defining number of regular graphs. Let \(d(n,r, \chi = k)\) be the smallest defining number of all \(r\)-regular \(k\)-chromatic graphs with \(n\) vertices, and \(f(n,k) = \frac{k-2}{2(k-1)} +\frac{2+(k-2)(k-3)}{2(k-1)}\). Mahmoodian and Mendelsohn (1999) determined the value of \(d(n,k, \chi = k)\) for all \(k \leq 5\), except for the case of \((n,k) = (10,5)\). They showed that \(d(n,k, \chi = k) = \lceil f(n,k) \rceil\), for \(k \leq 5\). They raised the following question: Is it true that for every \(k\), there exists \(n_0(k)\) such that for all \(n \geq n_0(k)\), we have \(d(n,k, \chi = k) = \lceil f(n,k) \rceil\)?
Here we determine the value of \(d(n,k, \chi = k)\) for each \(k\) in some congruence classes of \(n\). We show that the answer for the question above, in general, is negative. Also, for \(k = 6\) and \(k = 7\) the value of \(d(n,k, \chi = k)\) is determined except for one single case, and it is shown that \(d(10,5, \chi = 5) = 6\).
Let \((T_i)_{i\geq 0}\) be a sequence of trees such that \(T_{i+1}\) arises by deleting the \(b_i\) vertices of degree \(\leq 1\) from \(T_i\). We determine those trees of given degree sequence or maximum degree for which the sequence \(b_0, b_1, \ldots\) is maximum or minimum with respect to the dominance order. As a consequence, we also determine trees of given degree sequence or maximum degree that are of maximum or minimum Balaban index.
In this paper, we give a complete characterization of the pseudogracefulness of cycles.
The maximal clique that contains an edge which is not contained in any other maximal cliques is called essential. A graph in which each maximal clique is essential is said to be maximal clique irreducible. Maximal clique irreducible graphs were introduced and studied by W.D. Wallis and G.-H. Zhang in \(1990\) \([6]\). We extend the concept and define a graph to be weakly maximal clique irreducible if the set of all essential maximal cliques is a set of least number of maximal cliques that contains every edge. We characterized the graphs for which each induced subgraph is weakly maximal clique irreducible in \([4]\). In this article, we characterize the line graphs which are weakly maximal clique irreducible and also the line graphs which are maximal clique irreducible.
A weighted graph is one in which every edge \(e\) is assigned a non-negative number, called the weight of \(e\). For a vertex \(v\) of a weighted graph, \(d^w(v)\) is the sum of the weights of the edges incident with \(v\). For a subgraph \(H\) of a weighted graph \(G\), the weight of \(H\) is the sum of the weights of the edges belonging to \(H\). In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. Let \(G\) be a \(k\)-connected weighted graph where \(2 \leq k\). Then \(G\) contains either a Hamilton cycle or a cycle of weight at least \(2m/(k+1)\), if \(G\) satisfies the following conditions:(1)The weighted degree sum of any \(k\) independent vertices is at least \(m\),(2) \(w(xz) = w(yz)\) for every vertex \(z \in N(x) \cap N(y)\) with \(d(z,y) = 2\), and (3)In every triangle \(T\) of \(G\), either all edges of \(T\) have different weights or all edges of \(T\) have the same weight.
A circulant digraph \(G(a_1, a_2, \ldots, a_k)\), where \(0 < a_1 < a_2 < \ldots < a_k < |V(G)| = n\), is the vertex transitive directed graph that has vertices \(i+a_1, i+a_2, \ldots, i+a_k \pmod{n}\) adjacent to each vertex \(i\). We give the necessary and sufficient conditions for \(G(a_1, a_2)\) to be hamiltonian, and we prove that \(G(a, n-a, b)\) is hamiltonian. In addition, we identify the explicit hamiltonian circuits for a few special cases of sparse circulant digraphs.
We find a family of graphs each of which is not Hall \(t\)-chromatic for all \(t \geq 3\), and use this to prove that the same holds for the Kneser graphs \(K_{a,b}\) when \(a/b \geq 3\) and \(b\) is sufficiently large (depending on \(3 – (a/b)\)). We also make some progress on the problem of characterizing the graphs that are Hall \(t\)-chromatic for all \(t\).
The chromatic sum of \(G\), denoted by \(\sum(G)\), is the minimum sum of vertex colors, taken over all proper colorings of \(G\) using natural numbers. In general, finding \(\sum(G)\) is NP-complete. This paper presents polynomial-time algorithms for finding the chromatic sum for unicyclic graphs and for outerplanar graphs.
We enumerate all order ideals of a garland, a partially ordered set which generalizes crowns and fences. Moreover, we give some bijection between the set of such ideals and the set of certain kinds of lattice paths.
In this paper, we consider transformations between posets \(P\) and \(Q\), whose semi bound graphs are the same. Those posets with the same double canonical posets can be transformed into each other by a finite sequence of two kinds of transformations, called \(d\)-additions and \(d\)-deletions.
A paired-dominating set of a graph \(G\) is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\) is the minimum cardinality of a paired-dominating set of \(G\), and is obviously bounded below by the domination number of \(G\). We give a constructive characterization of the trees with equal domination and paired-domination numbers.
A recent series of papers by Anderson and Preece has looked at half-and-half terraces for cyclic groups of odd order, particularly focusing on those terraces which are narcissistic. We give a new direct product construction for half-and-half terraces which allows us to construct a narcissistic terrace for every abelian group of odd order. We also show that infinitely many non-abelian groups have narcissistic terraces.
Using generating functions of the author \(([1], [2])\), we obtain three infinite classes of combinatorial identities involving partitions with “\(n+t\) copies of \(n\)” introduced by the author and G.E. Andrews [3], and lattice paths studied by the author and D.M. Bressoud [4].
In this paper, we find necessary and sufficient conditions for the existence of a \(6\)-cycle system of \(K_n – E(R)\) for every \(2\)-regular, not necessarily spanning subgraph \(R\) of \(K_n\).
It is known that the smallest complete bipartite graph which is not \(3\)-choosable has \(14\) vertices. We show that the extremal configuration is unique.
We formalize the intuitive question of coloring the bricks of a wall in such a way that no repetition occurs in any row, nor any vertical line intersects two or more bricks with the same color. We achieve a complete classification up to the least number of required colors, among all dimensions of the walls, and all admitted incidences of the bricks. The involved combinatorial structures (namely, \(regular\) \(walls\)) are a special case of more general structures, which can be interpreted as adjacency matrices of suitable directed hypergraphs. Coloring the bricks is equivalent to coloring the arcs of the corresponding hypergraph. Regular walls seem interesting also for their connections with latin rectangles.
Tutte’s \(3\)-flow conjecture is equivalent to the assertion that there exists an orientation of the edges of a \(4\)-edge-connected, \(5\)-regular graph \(G\)for which the out-flow at each vertex is \(+3\) or \(-3\). The existence of one such orientation of the edges implies the existence of an equipartition of the vertices of \(G\) that separates the two possible types of vertices. Such an equipartition is called mod \(3\)-orientable. We give necessary and sufficient conditions for the existence of mod \(3\)-orientable equipartitions in general \(5\)-regular graphs, in terms of:(i) a perfect matching of a bipartite graph derived from the equipartition;(ii) the sizes of cuts in \(G\).Also, we give a polynomial-time algorithm for testing whether an equipartition of a \(5\)-regular graph is mod \(3\)-orientable.
In this paper, we look at generalizations of Stirling numbers which arise for arbitrary integer sequences and their \(k\)-th powers. This can be seen as a complementary strategy to the unified approach suggested in [9]. The investigations of [3] and [14] present a more algebraically oriented approach to generalized Stirling numbers.
In the first and second sections of the paper, we give the corresponding formulas for the generalized Stirling numbers of the second and first kind, respectively. In the third section, we briefly discuss some examples and special cases, and in the last section, we apply the square case to facilitate a counting approach for set partitions of even size.
In this paper, we give two sufficient conditions for a graph to be type \(1\) with respect to the total chromatic number and prove the following results:
(i) If \(G\) and \(H\) are of type \(1\), then \(G \times H\) is of type \(1\);
(ii) If \(\varepsilon(G) \leq v(G) + \frac{3}{2}\Delta(G) – 4\), then \(G\) is of type \(1\).
We prove several results dealing with various counting functions for partitions of an integer into four squares of equal parity. Some are easy consequences of earlier work, but two are new and surprising. That is, we show that the number of partitions of \(72n+ 60\) into four odd squares (distinct or not) is even.
We prove that if \(G\) is a simple graph of order \(n \geq 3k\) such that \(|N(x) \cup N(y)| \geq 3k\) for all nonadjacent pairs of vertices \(x\) and \(y\), then \(G\) contains \(k\) vertex-independent cycles.
The non-planar vertex deletion or vertex deletion \(vd(G)\) of a graph \(G = (V, E)\) is the smallest non-negative integer \(k\) such that the removal of \(k\) vertices from \(G\) produces a planar graph. Hence, the maximum planar induced subgraph of \(G\) has precisely \(|V| – vd(G)\) vertices. The problem of computing vertex deletion is in general very hard; it is NP-complete. In this paper, we compute the non-planar vertex deletion for the family of toroidal graphs \(C_n \times C_m\).
The graph resulting from contracting edge \( e \) is denoted \( G/e \). An edge \( e \) is radius-essential if \( rad(G/e) < rad(G) \). Let \( c_r(G) \) denote the number of radius-essential edges in graph \( G \). In this paper, we study realizability questions relating to the number of radius-essential edges, give bounds on \( c_r(G) \) in terms of radius and order, and we characterize various classes of graphs achieving extreme values of \( c_r(G) \).
We prove tight estimates on the minimum weight of an edge decomposition of the complete graph into subgraphs of 3 or 4 edges, where the weight of a subgraph is the number of its vertices. We conjecture that the weighted edge decomposition problem on general graphs is NP-complete for every \( k > 2 \). This conjecture is shown to be true for every \( k \leq 11 \) except \( k = 8 \). The problem is motivated by the traffic grooming problem for optical networks.
There are six distinct ways in which the vertices of a 4-cycle may be coloured with two colours, called \(\text{colouring types}\). Let \( C \) be the set of these colouring types and let \( S \) be a non-empty subset of \( C \). Suppose we colour the vertices of \( K_v \) with two colours. If \( D \) is a 4-cycle decomposition of \( K_v \) such that the colouring type of each 4-cycle is in \( S \), then \( D \) is said to have a \({colouring\; of\; type}\) \( S \). Furthermore, the colouring is said to be \({proper}\) if every colouring type in \( S \) is represented in \( D \). For all possible \( S \) of size one, two or three, excluding three cases already settled, we completely settle the existence question for 4-cycle decompositions of \( K_v \) with a colouring of type \( S \).
For a solution \( S \) of the \( n \)-queens problem, let \( M(S) \) denote the maximum of the absolute values of the diagonal numbers of \( S \), and let \( m(S) \) denote the minimum of those absolute values. For \( n \geq 4 \), let \( F(n) \) denote the minimum value of \( M(S) \), and let \( f(n) \) denote the maximum value of \( m(S) \), as \( S \) ranges over all solutions of the \( n \)-queens problem. Say that a solution \( S \) is an \( n \)-\({champion}\) if \( M(S) = F(n) \) and \( m(S) = f(n) \).
Approximately linear bounds are given for \( F(n) \) and \( f(n) \), along with computational results and several constructions together providing evidence that the bounds are excellent. It is shown that, in the range \( 4 \leq n \leq 24 \), \( n \)-champions exist except for \( n = 11, 16, 21, 22 \).
Let \( S \) be a stable set in a graph \( G \), possibly \( S = \emptyset \). The subgraph \( G – N[S] \), where \( N[S] \) is the closed neighborhood of \( S \), is called a \({co-stable \;subgraph}\) of \( G \). We denote by \( \text{CSub}(G) \) the set of all co-stable subgraphs of \( G \). A class of graphs \( \mathcal{P} \) is called \({co-hereditary}\) if \( G \in \mathcal{P} \) implies \( \text{CSub}(G) \subseteq \mathcal{P} \). Our result: If the set of all minimal forbidden co-stable subgraphs for a non-empty co-hereditary class \( \mathcal{P} \) is finite, then Stable Set is an NP-complete problem within \( \mathcal{P} \). Also, we prove that the decision problem of recognizing whether a graph has a fixed graph \( H \) as a co-stable subgraph is NP-complete for each non-trivial graph \( H \).
In this paper, we show the cordiality of the following families of graphs: (1) Pyramid graphs, (2) One point unions of plys,(3) One point unions of wheel related graphs, (4) Path unions of shells of different sizes, (5) Path unions of flags of different sizes.
Many different approaches exist in studying graphs with high connectivity and small diameter. We consider the effect of deleting vertices and edges from a graph while maintaining a small diameter. The following property is introduced: A graph \( G \) has property \( B_{d,i,j} \) if and only if after the removal of at most \( i \) vertices and at most \( j \) edges, the resulting graph has diameter at most \( d \) and is not the trivial graph on one vertex. The central theme of this paper is to investigate the structure of graphs that have property \( B_{d,i,j} \) and to investigate the structure that is needed to imply that a graph has property \( B_{d,i,j} \). Lower bounds on minimum degree and connectivity that imply property \( B_{d,i,j} \) for specific values of \( d \) are found. These bounds are also shown to be sharp in all but one case.
An \( m \)-cycle system of order \( v \), denoted by \( mCS(v) \), is a decomposition of the complete graph \( K_v \) into \( m \)-cycles. We discuss two types of large sets of \( mCS(v) \) and construct examples of both types for \( (m,v) = (4,9) \) and one type for \( (m,v) = (6,9) \). These are the first large sets of cycle systems constructed with \( m > 3 \), apart from the Hamiltonian cycle decompositions given in [2].
For two vertices \( u \) and \( v \) in a connected graph \( G \), the detour distance \( D(u,v) \) from \( u \) to \( v \) is defined as the length of a longest \( u-v \) path in \( G \). The detour eccentricity \( e_D(v) \) of a vertex \( v \) in \( G \) is the maximum detour distance from \( v \) to a vertex of \( G \). The detour radius \( \text{rad}_D(G) \) of \( G \) is the minimum detour eccentricity among the vertices of \( G \), while the detour diameter \( \text{diam}_D(G) \) of \( G \) is the maximum detour eccentricity among the vertices of \( G \). It is shown that \(\text{rad}_D(G) < \text{diam}_D(G) < 2\text{rad}_D(G)\) for every connected graph \( G \) and that every pair \( a,b \) of positive integers with \( a \leq b \leq 2a \) is realizable as the detour radius and detour diameter of some connected graph. The detour center of \( G \) is the subgraph induced by those vertices of \( G \) having detour eccentricity \( \text{rad}_D(G) \). A connected graph \( G \) is detour self-centered if \( G \) is its own detour center. The detour periphery of \( G \) is the subgraph induced by the vertices of \( G \) having detour eccentricity \( \text{diam}_D(G) \). It is shown that every graph is the detour center of some connected graph. Detour self-centered graphs are investigated. We present sufficient conditions for a graph to be the detour periphery of some connected graph. Several classes of graphs that are not the detour periphery of any connected graph are determined.
In recent work, Corteel and Lovejoy extensively studied overpartitions as a means of better understanding and interpreting various \( q \)-series identities. Our goal in this article is quite different. We wish to prove a number of arithmetic relations satisfied by the overpartition function. Employing elementary generating function dissection techniques, we will prove identities such as
\[
\sum\limits_{n\geq0}\overline{p}\left(8n + 7\right) q^n = 64 \frac{(q^2)_\infty^{22}}{(q)_\infty^{23}}
\]
and congruences such as
\[
\overline{p}(9n+6) \equiv 0 \pmod{8}
\]
where \( \overline{p}(n) \) denotes the number of overpartitions of \( n \).
Let \( G = (V,E) \) be a graph with \( |V| = p \) and \( |E| = q \). The graph \( G \) is total edge-magic if there exists a bijection \( f : V \cup E \to \{1,2,\ldots,p+q\} \) such that for all \( e = (u,v) \in E \), \( f(u) + f(e) + f(v) \) is constant throughout the graph. A total edge-magic graph is called super edge-magic if \( f(V) = \{1,2,\ldots,p\} \). Lee and Kong conjectured that for any odd positive integer \( r \), the union of any \( r \) star graphs is super edge-magic. In this paper, we supply substantial new evidence to support this conjecture for the case \( r = 3 \).
We show that \( \mathbb{Z} \)-cyclic ordered triplewhist and directed triplewhist tournaments on \( p \) elements exist when \( p \equiv 9 \pmod{16} \) is prime.
If \( G = (V,E,F) \) is a finite connected plane graph on \( |V| = p \) vertices, \( |E| = q \) edges and \( |F| = t \) faces, then \( G \) is said to be \( (a, d) \)-face antimagic iff there exists a bijection \( h: E \to \{1,2,\ldots,q\} \) and two positive integers \( a \) and \( d \) such that the induced mapping \( g_h: F \to \mathbb{N} \), defined by \( g_h(f) = \sum\{h(u,v) : \text{edge } (u,v) \text{ surrounds the face } f\} \), is injective and has the image set \( g_h(F) = \{a,a+d,\ldots,a + (t – 1)d\} \). We deal with \( (a,d) \)-face antimagic labelings for a certain class of plane graphs.
We provide tables which summarize various aspects of the finite linear groups \( \text{GL}(n, 2) \), \( n < 7 \), in their action upon the vector space \( V_n = V(n, 2) \) and upon the associated projective space \( \text{PG}(n – 1, 2) \). It is intended that the tabulated results should be immediately accessible to finite geometers, and to all others (design theorists, coding theorists, \ldots) who have occasional need of these groups. In the case \( n = 4 \) attention is also paid to the maximal subgroup \( \Gamma \text{L}(2, 4) \). In the case \( n = 6 \) the maximal subgroups \( \Gamma \text{L}(2, 8) \) and \( \Gamma \text{L}(3, 4) \) are treated, as are class aspects of the tensor product structure \( V_6 = V_2 \otimes V_3 \), and of the exterior product structure \( V_6 = \wedge^2 V_4 \).
It is conjectured that any 2-regular graph \( G \) with \( n \) edges has a \( \rho \)-labeling (and thus divides \( K_{2n+1} \) cyclically). In this note, we show that the conjecture holds when \( G \) has at most two components.
Let \(K_4\backslash e=…\). If we remove the “diagonal” edge, the result is a \(4\)-cycle. Let \((X,B)\) be a \(K_4\backslash e\) design of order \(n\); i.e., an edge-disjoint decomposition of \(K_n\) into copies of \(K_4\backslash e\). Let \(D(B)\) be the collection of “diagonals” removed from the graphs in \(B\) and \(C(B)\) the resulting collection of \(4\)-cycles. If \(C_2(B)\) is a reassembly of these edges into \(4\)-cycles and \(L\) is the collection of edges in \(D(B)\) not used in a \(4\)-cycle of \(C_2(B)\), then \((X, (C_1(B) \cup C_2(B)), L)\) is a packing of \(K_n\) with \(4\)-cycles and is called a metamorphosis of \((X,B)\). We construct, for every \(n = 0\) or \(1\) (mod \(5\)) \(> 6\), \(n \neq 11\), a \(K_4\backslash e\) design of order \(n\) having a metamorphosis into a maximum packing of \(K_n\) with \(4\)-cycles. There exists a maximum packing of \(K_n\) with \(4\)-cycles, but it cannot be obtained from a \(K_4\backslash e\) design.
We investigate the supereulerian graph problems within planar graphs, and we prove that if a \(2\)-edge-connected planar graph \(G\) is at most three edges short of having two edge-disjoint spanning trees, then \(G\) is supereulerian except for a few classes of graphs. This is applied to show the existence of spanning Eulerian subgraphs in planar graphs with small edge cut conditions. We also determine several extremal bounds for planar graphs to be supereulerian.
Given an acyclic digraph \(D\), the phylogeny graph \(P(D)\) is defined to be the undirected graph with \(V(D)\) as its vertex set and with adjacencies as follows: two vertices \(x\) and \(y\) are adjacent if one of the arcs \((x,y)\) or \((y,x)\) is present in \(D\), or if there exists another vertex \(z\) such that the arcs \((x,z)\) and \((y,z)\) are both present in \(D\). Phylogeny graphs were introduced by Roberts and Sheng [6] from an idealized model for reconstructing phylogenetic trees in molecular biology, and are closely related to the widely studied competition graphs. The phylogeny number \(p(G)\) for an undirected graph \(G\) is the least number \(r\) such that there exists an acyclic digraph \(D\) on \(|V(G)| + r\) vertices where \(G\) is an induced subgraph of \(P(D)\). We present an elimination procedure for the phylogeny number analogous to the elimination procedure of Kim and Roberts [2] for the competition number arising in the study of competition graphs. We show that our elimination procedure computes the phylogeny number exactly for so-called “kite-free” graphs. The methods employed also provide a simpler proof of Kim and Roberts’ theorem on the exactness of their elimination procedure for the competition number on kite-free graphs.
A multiple shell \(MS\{n_1^{t_1}, n_2^{t_2}, \dots, n_r^{t_r}\}\) is a graph formed by \(t_i\) shells of widths \(n_i\), \(1 \leq i \leq r\), which have a common apex. This graph has \(\sum_{i=1}^rt_i(n_i-1) + 1\) vertices. A multiple shell is said to be balanced with width \(w\) if it is of the form \(MS\{w^t\}\) or \(MS\{w^t, (w+1)^s\}\). Deb and Limaye have conjectured that all multiple shells are harmonious, and shown that the conjecture is true for the balanced double shells and balanced triple shells. In this paper, the conjecture is proved to be true for the balanced quadruple shells.
In [BabStein], Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In \([Kit1]\), Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There, either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper, we find the exponential generating function for the remaining case. Also, we consider permutations that avoid a pattern of the form \(x – yz\) or \(xy – z\) and begin with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(23\ldots 1k\), \((k-1)(k-2)\ldots 1k\), or end with one of the patterns \(12\ldots k\), \(k(k-1)\ldots 1\), \(1k(k-1)\ldots 2\), \(k12\ldots (k-1)\). For each of these cases, we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides, we generalize some of the obtained results as well as some of the results given in \([Kit3]\): we consider permutations avoiding certain generalized \(3\)-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.
In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (\(LEE\)) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of \(LEE\)-graphs over the set of all connected spanning \(k\)-edge subgraphs of a connected graph \(G\), and show that both of them are connected for a \(2\)-connected graph \(G\).
We look at binary strings of length \(n\) which contain no odd run of zeros and express the total number of such strings, the number of zeros, the number of ones, the total number of runs, and the number of levels, rises, and drops as functions of the Fibonacci and Lucas numbers and also give their generating functions. Furthermore, we look at the decimal value of the sum of all binary strings of length \(n\) without odd runs of zeros considered as base \(2\) representations of decimal numbers, which interestingly enough are congruent (mod \(3\)) to either \(0\) or a particular Fibonacci number. We investigate the same questions for palindromic binary strings with no odd runs of zeros and obtain similar results, which generally have different forms for odd and even values of \(n\).
In this paper, we characterize the potentially \(K_4\)-graphic sequences. This characterization implies the value \(\sigma(K_4,n)\), which was conjectured by P. Erdős, M. S. Jacobson, and J. Lehel [1] and was confirmed by R. J. Gould, M. S. Jacobson, and J. Lehel [2] and Jiong-Sheng Li and Zixia Song [5], independently.
The graph …….is called a kite and the decomposition of \(K_n\) into kites is called a kite system. Such systems exist precisely when \(n = 0\) or \(1\) (mod \(8\)). In \(1975\), C. C. Lindner and A. Rosa solved the intersection problem for Steiner triple systems. The object of this paper is to give a complete solution to the triangle intersection problem for kite systems (\(=\) how many triangles can two kite systems of order \(n\) have in common). We show that if \(x \in \{0, 1, 2, \dots, n(n-1)/8\}\), then there exists a pair of kite systems of order \(n\) having exactly \(n\) triangles in common.
A directed balanced incomplete block design (\(DB\)(\(k\), \(\lambda\);\(v\))) \((X, \mathcal{B})\) is called self-converse if there is an isomorphic mapping \(f\) from \((X, \mathcal{B})\) to \((X, \mathcal{B}^{-1})\), where \(\mathcal{B}^{-1} = \{B^{-1} : B \in \mathcal{B}\}\) and \(B^{-1} = (x_k,x_{k-1},\ldots,x_2,x_1)\) for \(B=(x_1,x_2,\ldots,x_{k-1},x_{k})\). In this paper, we give the existence spectrum for self-converse \(DB\)(\(4\),\(\lambda\);\(v\)) for any \(\lambda \geq 1\).
A sequence \(\pi = (d_1, \dots, d_n)\) of nonnegative integers is graphic if there exists a graph \(G\) with \(n\) vertices for which \(d_1, \dots, d_n\) are the degrees of its vertices. \(G\) is referred to as a realization of \(\pi\). Let \(P\) be a graph property. A graphic sequence \(\pi\) is potentially \(P\)-graphic if there exists a realization of \(\pi\) with the graph property \(P\). Similarly, \(\pi\) is forcibly \(P\)-graphic if all realizations of \(\pi\) have the property \(P\). We characterize potentially Halin graph-graphic sequences, forcibly Halin graph-graphic sequences, and forcibly cograph-graphic sequences.
We establish the nonexistence of:(i) Steiner \(t\)-\((v,k)\) trades of volume \(s\), for \(2^t + 2^{t-1} < s t+1\) and volume \(s < (t-1)2^t + 2\).
Using R. C. Read’s superposition method, we establish a formula for the enumeration of Euler multigraphs, with loops allowed and with given numbers of edges. In addition, applying Burnside’s Lemma and our adaptation of Read’s superposition method, we also derive a formula for the enumeration of Euler multigraphs without loops — via the calculation of the number of perfect matchings of the complement of complete multipartite graphs. MAPLE is employed to implement these enumerations. For one up to \(13\) edges, the numbers of nonisomorphic Euler multigraphs with loops allowed are:\(1, 3, 6, 16, 34, 90, 213, 572, 1499, 4231, 12115, 36660, 114105\) respectively, and for one up to \(16\) edges, the numbers of nonisomorphic Euler multigraphs without loops are:\(0, 1, 1, 4, 4, 15, 22, 68, 131, 376, 892, 2627, 7217, 22349, 69271, 229553\) respectively. Simplification of these methods yields the numbers of multigraphs with given numbers of edges, results which also appear to be new. Our methods also apply to multigraphs with essentially arbitrary constraints on vertex degrees.
In this paper, we determine the number of all maximal \(k\)-independent sets in the generalized lexicographical product of graphs. We construct a polynomial that calculates this number using the concept of Fibonacci polynomials and generalized Fibonacci polynomials. Also, for special graphs, we give the recurrence formula.
For a \(3\)-vertex coloring, a face of a triangulation whose vertices receive all three colors is called a vivid face with respect to it. In this paper, we show that for any triangulation \(G\) with \(n\) faces, there exists a coloring of \(G\) with at least \( \frac{1}{2}n\) faces and construct an infinite series of plane triangulations such that any \(3\)-coloring admits at most \(\frac{1}{5}(3n- 2)\) vivid faces.
A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes \(O(n^2)\) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions.
One application for this work is graph visualization. Projective plane embeddings can be represented on the plane and can provide aesthetically pleasing pictures of some non-planar graphs. More important is that it is highly likely that many problems that are computationally intractable (for example, NP-complete or #P-complete) have polynomial time algorithms when restricted to graphs of fixed orientable or non-orientable genus. Embedding the graph on the surface is likely to be the first step for these algorithms.
We consider the nonexistence of \(e\)-perfect codes in the Johnson scheme \(J(n, w)\). It is proved that for each \(J(2w + 3p, w)\) for \(p\) prime and \(p \neq 2, 5\), \(J(2w + 5p, w)\) for \(p\) prime and \(p \neq 3\), and \(J(2w + p^2, w)\) for \(p\) prime, it does not contain non-trivial \(e\)-perfect codes.
A graph \(G\) is called \(f\)-factor-covered if every edge of \(G\) is contained in some \(f\)-factor. \(G\) is called \(f\)-factor-deleted if \(G\) – \(e\) contains an \(f\)-factor for every edge \(e\). Babler proved that every \(r\)-regular, \((r – 1)\)-edge-connected graph of even order has a \(1\)-factor. In the present article, we prove that every \(2r\)-regular graph of odd order is both \(2m\)-factor-covered and \(2m\)-factor-deleted for all integers \(m\), \(1 \leq m \leq r – 1\), and every \(r\)-regular, \((r – 1)\)-edge-connected graph of even order is both \(m\)-factor-covered and \(m\)-factor-deleted for all integers \(m\), \(1 \leq m \leq \left\lfloor \frac{r}{2} \right\rfloor\).
The convex hull of a subset \(A\) of \(V(G)\), where \(G\) is a connected graph, is defined as the smallest convex set in \(G\) containing \(A\). The hull number of \(G\) is the cardinality of a smallest set \(A\) whose convex hull is \(V(G)\). In this paper, we give the hull number of the composition of two connected graphs.
The basis number \(b(G)\) of a graph \(G\) is defined to be the least integer \(d\) such that \(G\) has a \(d\)-fold basis for its cycle space. In this paper, we investigate the basis number of the direct product of theta graphs and paths.
Large sets of balanced incomplete block (\(BIB\)) designs and resolvable \(BIB\) designs are discussed. Some recursive constructions of such large sets are given. Some existence results, in particular for practical \(k\), are reviewed.
We consider point-line geometries having three points on every line, having three lines through every point (\(bi\)-\(slim\; geometries\)), and containing triangles. We give some (new) constructions and we prove that every flag-transitive such geometry either belongs to a certain infinite class described by Coxeter a long time ago, or is one of three well-defined sporadic ones, namely, The Möbius-Kantor geometry on \(8\) points, The Desargues geometry on \(10\) points,A unique infinite example related to the tiling of the real Euclidean plane in regular hexagons.We also classify the possible groups.
Let \(G\) be a simple graph such that \(\delta(G) \geq \lfloor\frac{|V(G)|}{2}\rfloor + k\), where \(k\) is a non-negative integer, and let \(f: V(G) \to \mathbb{Z}^+\) be a function having the following properties (i)\(\frac{d_G(x)}{2}-\frac{k+1}{2}\leq f(x)\leq \frac{d_G(x)}{2}+\frac{k+1}{2}\) for every \(x \in V(G)\), (ii)\(\sum\limits_{x\in V(G)}f(x)=|E(G)|\). Then \(G\) has an orientation \(D\) such that \(d^+_D(x) = f(x)\), for every \(x \in V(G)\).
The so-called multi-restricted numbers generalize and extend the role of Stirling numbers and Bessel numbers in various problems of combinatorial enumeration. Multi-restricted numbers of the second kind count set partitions with a given number of parts, none of whose cardinalities may exceed a fixed threshold or “restriction”. The numbers are shown to satisfy a three-term recurrence relation. Both analytic and combinatorial proofs for this relation are presented. Multi-restricted numbers of both the first and second kinds provide connections between the orbit decompositions of subsets of powers of a finite group permutation representation, in which the number of occurrences of elements is restricted. An exponential generating function for the number of orbits on such restricted powers is given in terms of powers of partial sums of the exponential function.
A class of graphs called generalized ladder graphs is defined. A sufficient condition for pairs of these graphs to be chromatically equivalent is proven. In addition, a formula for the chromatic polynomial of a graph of this type is proven. Finally, the chromatic polynomials of special cases of these graphs are explicitly computed.
Let \(k \geq 3\) be odd and \(G = (V(G), E(G))\) be a \(k\)-edge-connected graph. For \(X \subseteq V(G)\), \(e(X)\) denotes the number of edges between \(X\) and \(V(G) – X\). We here prove that if \(\{s_i, t_i\} \subseteq X_i \subseteq V(G)\), \(i = 1, 2\), \(X_1 \cap X_2 = \emptyset\), \(e(X_1) \leq 2k-2\) and \(e(X_2) < 2k-1\), then there exist paths \(P_1\) and \(P_2\) such that \(P_i\) joins \(s_i\) and \(t_i\), \(V(P_i) \subseteq X_i\) (\(i = 1, 2\)) and \(G – E(P_1 \cup P_2)\) is \((k-2)\)-edge-connected. And in fact, we give a generalization of this result and some other results about paths not containing given edges.
Optimal binary linear codes of length \(18\) containing the \([6, 5, 2]\otimes[ 3, 1, 3]\) product code are presented. It is shown that these are \([18, 9, 5]\) and \([18, 8, 6]\) codes. The soft-decision maximum-likelihood decoding complexity of these codes is determined. From this point of view, these codes are better than the \([18, 9, 6]\) code.
An elongated ply \( T(n; t^{(1)}, t^{(2)}, \ldots, t^{(n)}) \) is a snake of \( n \) number of plys \( P_{t(i)} (u_i, u_{i+1}) \) where any two adjacent plys \( P_{t(i)} \) and \( P_{t(i+1)} \) have only the vertex \( u_{i+1} \) in common. That means the block cut vertex graph of \( T_n \) is thus a path of length \( n – 1 \). In this paper, the cordiality of the Elongated Ply \( T_n \) is investigated.
Consider placing a guard on each vertex of a dominating set \( S_0 \) of a graph. If for every vertex \( v \notin S_0 \), there is a corresponding guard at an adjacent vertex \( u \) for which the resulting set \( S_1 = S_0 – \{u\} \cup \{v\} \) is dominating, then we say that \( S_0 \) is \( 1 \)-secure. It is eternally \( 1 \)-secure if for 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 \( u_i \) equal to or adjacent to \( v_i \), such that each set \( S_i = S_{i-1} – \{u_i\} \cup \{v_i\} \) is dominating. We investigate the minimum cardinality of an eternally secure set. In particular, we refute a conjecture of Burger et al. We also investigate eternal \( m \)-security, in which all guards can move simultaneously.
In 1990, Kolesova, Lam, and Thiel determined the 283,657 main classes of Latin squares of order 8. Using techniques to determine relevant Latin trades and integer programming, we examine representatives of each of these main classes and determine that none can contain a uniquely completable set of size less than 16. In three of these main classes, the use of trades which contain less than or equal to three rows, columns, or elements does not suffice to determine this fact. We closely examine properties of representatives of these three main classes. Writing the main result in Nelder’s notation for critical sets, we prove that \( \text{scs}(8) = 16 \).
An edge-ordering of a graph \( G = (V, E) \) is a one-to-one function \( f \) from \( E \) to the set of positive integers. A path of length \( k \) in \( G \) is called a \( (k, f) \)-ascent if \( f \) increases along the edge sequence of the path. The altitude \( \alpha(G) \) of \( G \) is the greatest integer \( k \) such that for all edge-orderings \( f \), \( G \) has a \( (k, f) \)-ascent.
We obtain a recursive lower bound for \( \alpha(K_{m,n}) \) and show that
\[\alpha(K_{3,n}) = \begin{cases}4 & \text{if } 5 \leq n \leq 9 \\5 & \text{if } 10 \leq n \leq 12 \\6 & \text{if } n \geq 13\end{cases}\]
A vertex set \( D \) of a graph \( G \) is a dominating set if every vertex not in \( D \) is adjacent to some vertex in \( D \). The domination number \( \gamma \) of a graph \( G \) is the minimum cardinality of a dominating set in \( G \). In 1989, Brigham and Dutton [1] proved
\[\gamma \leq \left\lceil\frac{3n-g}{6}\right\rceil\]
for each graph \( G \) of order \( n \), minimum degree \( \delta \geq 2 \), and girth \( g \geq 5 \). If \( G \) is a graph of order \( n \), minimum degree \( \delta \geq 2 \), girth \( g \geq 5 \) and neither a cycle nor one of two exceptional graphs, then we give in this paper the better bound
\[\gamma(G) \leq \left\lceil\frac{3n-g}{6}\right\rceil-1\]
For \( \delta \geq 3 \) and \( g \geq 5 \), we also prove \( \gamma \leq \left\lceil\frac{6n-g}{15}\right\rceil \), and this inequality is better than \( (*) \) when \( n > g + 10 \). In addition, if \( \delta \geq 3 \), then we show that
\[2\gamma \leq n – (\delta-2)(1 + \lfloor{d}/{3}\rfloor)\]
where \( d \) is the diameter of the graph. Some related bounds in terms of the diameter, girth, order, and minimum degree are also presented.
SOS-skeins correspond exactly to the Steiner quadruple systems [8,12]. Let \( P_1 \) be a finite simple SQS-skein of cardinality \( n > 4 \). In this article, we will present a construction for a non-simple subdirectly irreducible (monolithic) SOS-skein \( P = 2 \otimes_\alpha P_n \) of cardinality \( 2n \) in which each proper homomorphic image is Boolean for all \( n \equiv 2 \) or \( 4 \pmod{6} \). We can then show that if \( P_1 \) has a simple derived sloop, then the constructed SOS-skein \( 2 \otimes_\alpha P_1 \) contains a derived sloop which is subdirectly irreducible and has the same property as the SOS-skein \( 2 \otimes_\alpha P_1 \) that each of its proper homomorphic images is Boolean. Similar to the theory of Steiner loops and Steiner quasigroups [14], the author [1] has proven that the variety \( V(P_1) \) generated by a finite simple cubic SQS-skein \( P_1 \) covers the smallest non-trivial subvariety (the class of all Boolean SQS-skeins). Finally, we show that the variety \( V(2 \otimes_\alpha P_1) \) generated by the constructed SQS-skein \( 2 \otimes_\alpha P_1 \) covers the variety \( V(P_1) \) for each finite simple cubic SOS-skein \( P_1 \).
It is shown that for any rank \( r \) with \( n – \log(n+1) + 4 \leq r \leq n – 4 \) and any length \( n \), where \( n = 2^k – 1 \) and \( k \geq 8 \), there is a perfect code with these parameters and with a trivial group of symmetries.
In this paper, we introduce, for the first time, the notion of self-dual modular-graceful labeling of a cyclic digraph. A cyclic digraph \( G(V, E) \) is a digraph whose connected components are directed cycles. The line digraph \( G^\wedge(V^\wedge, E^\wedge) \) of the cyclic digraph \( G \) is the digraph where \( V^\wedge = E \), \( E^\wedge = V \), and if \( \alpha, \beta \) are two edges of \( G \) which join vertex \( x \) to vertex \( y \) and vertex \( y \) to vertex \( z \) respectively, then in the digraph \( G^\wedge \), \( y \) is the edge joining vertex \( \alpha \) to vertex \( \beta \). A labeling \( f \) for a cyclic digraph of order \( n \) is a map from \( V \) to \( \mathbb{Z}_{n+1} \). The labeling \( f \) induces a dual labeling \( f^\wedge \) for \( G^\wedge \) by \( f^\wedge(\alpha) = f(x) – f(y) \), where \( \alpha \) is an edge of \( G \) which joins vertex \( x \) to vertex \( y \). A self-dual modular-graceful cyclic digraph \( G \) is a cyclic digraph together with a labeling \( f \) where the image \( f(V) = \mathbb{Z}_{n+1}^* \), and \( \langle G^\wedge, f^\wedge \rangle \) is an isomorphic digraph of \( \langle G, f \rangle \). We prove the necessary and sufficient conditions for the existence of self-dual modular-graceful cyclic digraphs and connected self-dual modular-graceful cyclic digraphs. We also give some explicit constructions of these digraphs in the case \( n+1 \) is prime and in the general case where \( n+1 \) is not prime.
We refer to a labeling of a plane graph as a \( d \)-antimagic labeling if the vertices, edges, and faces of the graph are labeled in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face and the weights of all the faces form an arithmetic progression of difference \( d \). This paper describes \( d \)-antimagic labelings for a special class of plane graphs.
2-trees are defined recursively, starting from a single edge, by repeatedly erecting new triangles onto existing edges. These have been widely studied in connection with chordal graphs, series-parallel graphs, and isolated failure immune (IFI) networks.
A similar family, based on recursively erecting new \( K_{2,h} \) subgraphs onto existing edges, is shown to have analogous connections to chordal bipartite graphs, series-parallel graphs, and a notion motivated by IFI networks.
A graceful labeling of a graph \( G \) of size \( n \) is an assignment of labels from \( \{0, 1, \ldots, n\} \) to the vertices of \( G \) such that when each edge has assigned a weight defined by the absolute difference of its end-vertices, the resulting weights are distinct. The gracefulness of a graph \( G \) is the smallest positive integer \( k \) for which it is possible to label the vertices of \( G \) with distinct elements from the set \( \{0, 1, \ldots, k\} \) in such a way that distinct edges have distinct weights. In this paper, we determine the gracefulness of the union of cycles and complete bipartite graphs. We also give graceful labelings of unions of complete bipartite graphs.
The expected value and the variance of a multiplicity of a given part size in a random composition of an integer is obtained. This result was used in [1] to analyze algorithms for computing the Walsh-Hadamard transform.
We enumerate the balanced tournament designs on 10 points (BTD(5)) and find that there are exactly 30,220,557 nonisomorphic designs. We also find that there are exactly two nonisomorphic partitioned BTD(5)’s and 8,081,114 factored BTD(5)’s on 10 points. We enumerate other classes of balanced tournament designs on 10 points and give examples of some of the more interesting ones. In 1988, Corriveau enumerated the nonisomorphic BTD(4)’s, finding that there are 47 of them. This paper enumerates the next case and provides another good example of the combinatorial explosion phenomenon.
We examine decompositions of complete graphs with an even number of vertices into isomorphic spanning trees. We develop a cyclic factorization of \( K_{2n} \) into non-symmetric spanning trees. Our factorization methods are based on flexible \( q \)-labeling and blended labeling, introduced by Froncek. In this paper, we present several infinite classes of non-symmetric trees which have flexible \( q \)-labeling or blended labeling.
The analysis of the Tutte polynomial of a matroid using activities is associated with a shelling of the family of spanning sets. We introduce an activities analysis of the reliability of a system specified by an arbitrary clutter, associated with an \( \mathcal{S} \)-partition rather than a shelling. These activities are related to a method of constructing Boolean interval partitions developed by Dawson in the early 1980s.
Let \(D\) be a connected symmetric digraph, \(\Gamma\) a group of automorphisms of \(D\), and \(A\) a finite abelian group. For a cyclic \(A\)-cover of \(D\), we consider a lift of \(\gamma \in \Gamma\), and the associated group automorphism of some subgroup of \(Aut \, A\). Furthermore, we give a characterization for any \(\gamma \in \Gamma\) to have a lift in terms of some matrix.
It is shown that the voltage-current duality in topological graph theory can be obtained as a consequence of a combinatorial description of the pair (an embedded graph, the embedded dual graph)without any reference to derived graphs and derived embeddings. In the combinatorial description the oriented edges of an embedded graph are labeled by oriented edges of the embedded dual graph.
We extend the work of Currie and Fitzpatrick [1] on circular words avoiding patterns by showing that, for any positive integer \(n\), the Thue-Morse word contains a subword of length \(n\) which is circular cube-free. This proves a conjecture of V. Linek.
Let \(G\) be a simple graph with the average degree \(d_{ave}\) and the maximum degree \(\Delta\). It is proved, in this paper, that \(G\) is not critical if \(d_{ave} \leq \frac{103}{12}\) and \(\Delta \geq 12\). It also improves the current result by L.Y. Miao and J.L. Wu [7] on the number of edges of critical graphs for \(\Delta \geq 12\).
A \(3\)-restricted edge cut is an edge cut that disconnects a graph into at least two components each having order at least \(3\). The cardinality \(\lambda_3\) of minimum \(3\)-restricted edge cuts is called \(3\)-restricted edge connectivity. Let \(G\) be a connected \(k\)-regular graph of girth \(g(G) \geq 4\) and order at least \(6\). Then \(\lambda_3 \leq 3k – 4\). It is proved in this paper that if \(G\) is a vertex transitive graph then either \(\lambda_3 = 3k – 4\) or \(\lambda_3\) is a divisor of \(|G|\) such that \(2k – 2 \leq \lambda_3 \leq 3k – 5\) unless \(k = 3\) and \(g(G) = 4\). If \(k = 3\) and \(g(G) = 4\), then \(\lambda_3 = 4\). The extreme cases where \(\lambda_3 = 2k – 2\) and \(\lambda_3 = 3k – 5\) are also discussed.
Some classes of neighbour balanced designs in two-dimensional blocks are constructed. Some of these designs are statistically optimal and others are highly efficient when errors arising from units within each block are correlated.
Let \(G = (V, E)\) be a simple graph. For any real-valued function \(f: V \to {R}\) and \(S \subseteq V\), let \(f(S) = \sum_{v \in S} f(v)\). Let \(c, d\) be positive integers such that \(\gcd(c, d) = 1\) and \(0 < \frac{c}{d} \leq 1\). A \(\frac{c}{d}\)-dominating function (partial signed dominating function) is a function \(f: V \to \{-1, 1\}\) such that \(f(N[v]) \geq c\) for at least \(c\) of the vertices \(v \in V\). The \(\frac{c}{d}\)-domination number (partial signed domination number) of \(G\) is \(\gamma_{\frac{c}{d}}(G) = \min \{f(V) | f \text{ is a } \frac{c}{d}\text{-dominating function on } G\}\). In this paper, we obtain a few lower bounds of \(\gamma_{\frac{c}{d}}(G)\).
The groups \(G^{k,l,m}\) have been extensively studied by H. S. M. Coxeter. They are symmetric groups of the maps \(\{k,l\}_m\) which are constructed from the tessellations \(\{k,l\}\) of the hyperbolic plane by identifying two points, at a distance \(m\) apart, along a Petrie path. It is known that \(\text{PSL}(2,q)\) is a quotient group of the Coxeter groups \(G^{(m)}\) if \(-1\) is a quadratic residue in the Galois field \({F}_q\), where \(q\) is a prime power. G. Higman has posed the question that for which values of \(k,l,m\), all but finitely many alternating groups \(A_k\) and symmetric groups \(S_k\) are quotients of \(G^{k,l,m}\). In this paper, we have answered this question by showing that for \(k=3,l=11\), all but finitely many \(A_n\) and \(S_n\) are quotients of \(G^{3,11,m}\), where \(m\) has turned out to be \(924\).
The purpose of this article is to give combinatorial proofs of some binomial identities which were given by Z. Zhang.
Given \(t\geq 2\) cycles \(C_n\) of length \(n \geq 3\), each with a fixed vertex \(v^i_0\), \(i=1,2,\ldots,t\), let \(C^(t)_n\) denote the graph obtained from the union of the \(t\) cycles by identifying the \(t\) fixed vertices (\(v^1_0 = v^2_0 = \cdots = v^t_0\)). Koh et al. conjectured that \(C^(t)^n\) is graceful if and only if \(nt \equiv 0, 3 \pmod{4}\). The conjecture has been shown true for \(t = 3, 6, 4k\). In this paper, the conjecture is shown to be true for \(n = 5\).
Let \(G\) be a finite abelian group of exponent \(m\). By \(s(G)\) we denote the smallest integer \(c\) such that every sequence of \(t\) elements in \(G\) contains a zero-sum subsequence of length \(m\). Among other results, we prove that, let \(p\) be a prime, and let \(H = C_{p^{c_1}} \oplus \ldots C_{p^{c_l}}\) be a \(p\)-group. Suppose that \(1+\sum_{i=1}^{l}(p^{c_i}-1)=p^k\) for some positive integer \(k\). Then,\(4p^k – 3 \leq s(C_{p^k} \oplus H) \leq 4p^k – 2.\)
A connected dominating set \(D\) of a graph \(G\) has the property that not only does \(D\) dominate the graph but the subgraph induced by the vertices of \(D\) is also connected. We generalize this concept by allowing the subgraph induced by \(D\) to contain at most \(k\) components and examine the minimum possible order of such a set. In the case of trees, we provide lower and upper bounds and a characterization for those trees which achieve the former.
Let \(\sigma(K_{r,s}, n)\) denote the smallest even integer such that every \(n\)-term positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(K_{r,s}, n)\) has a realization \(G\) containing \(K_{r,s}\) as a subgraph, where \(K_{r,s}\) is the \(r \times s\) complete bipartite graph. In this paper, we determine \(\sigma(K_{2,3}, n)\) for \(m \geq 5\). In addition, we also determine the values \(\sigma(K_{2,s}, n)\) for \(s \geq 4\) and \(n \geq 2[\frac{(s+3)^2}{4}]+5\).
Formulas for vertex eccentricity and radius for the tensor product \(G \otimes H\) of two arbitrary graphs are derived. The center of \(G \otimes H\) is characterized as the union of three vertex sets of form \(A \times B\). This completes the work of Suh-Ryung Kim, who solved the case where one of the factors is bipartite. Kim’s result becomes a corollary of ours.
Let \(v, k,\lambda\) and \(n\) be positive integers. \((x_1, x_2, \ldots, x_k)\) is defined to be \(\{(x_i, x_j) : i \neq j, i,j =1,2,\ldots,k\},\) in which the ordered pair \((x_i, x_j)\) is called \((j-i)\)-apart for \(i > j\) and \((k+j-i)\)-apart for \(i > j\), and is called a cyclically ordered \(k\)-subset of \(\{x_1, x_2, \ldots, x_k\}\).
A perfect Mendelsohn design, denoted by \((v, k, \lambda)\)-PMD, is a pair \((X, B)\), where \(X\) is a \(v\)-set (of points), and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks), such that every ordered pair of points of \(X\) appears \(t\)-apart in exactly \(\lambda\) blocks of \(B\) for any \(t\), where \(1 \leq t \leq k-1\).
If the blocks of a \((v, k, \lambda)\)-PMD for which \(v \equiv 0 \pmod{k}\) can be partitioned into \(\lambda(v-1)\) sets each containing \(v/k\) blocks which are pairwise disjoint, the \((v, k, \lambda)\)-PMD is called resolvable, denoted by \((v, k, \lambda)\)-RPMD.
In the paper [14], we have showed that a \((v, 4, 1)\)-RPMD exists for all \(v \equiv 0 \pmod{4}\) except for \(4, 8\) and with at most \(49\) possible exceptions of which the largest is \(336\).
In this article, we shall show that a \((v, 4, 1)\)-RPMD for all \(v \equiv 0 \pmod{4}\) except for \(4, 8, 12\) and with at most \(27\) possible exceptions of which the largest is \(188\).
The independence number of Cartesian product graphs is considered. An upper bound is presented that covers all previously known upper bounds. A construction is described that produces a maximal independent set of a Cartesian product graph and turns out to be a reasonably good lower bound for the independence number. The construction defines an invariant of Cartesian product graphs that is compared with its independence number. Several exact independence numbers of products of bipartite graphs are also obtained.
Multi-loop digraphs are widely studied mainly because of their symmetric properties and their applications to loop networks. A multi-loop digraph, \(G = G(N; s_1, \ldots, s_\Delta)\) with \(1 \leq s_1 < \cdots < s_\Delta \leq N-1\) and \(\gcd(N, s_1, \ldots, s_\Delta) = 1\), has set of vertices \(V ={Z}_N\) and adjacencies given by \(v \mapsto v + s_i \mod N, i = 1, \ldots, \Delta\). For every fixed \(N\), an usual extremal problem is to find the minimum value \[D_\Delta(N)=\min\limits_{s_1,\ldots,s_\Delta \in Z_N}(N; s_1, \ldots, s_\Delta)\] where \(D(N; s_1, \ldots, s_\Delta)\) is the diameter of \(G\). A closely related problem is to find the maximum number of vertices for a fixed value of the diameter. For \(\Delta = 2\), all optimal families have been found by using a geometrical approach. For \(\Delta = 3\), only some dense families are known. In this work, a new dense family is given for \(\Delta = 3\) using a geometrical approach. This technique was already adopted in several papers for \(\Delta = 2\) (see for instance [5, 7]). This family improves the dense families recently found by several authors.
Gould et al. (Combinatorics, Graph Theory and Algorithms, Vol. 1 (1999), 387-400) considered a variation of the classical Turén-type extremal problems as follows: for a given graph \(H\), determine the smallest even integer \(\sigma (H,n)\) such that every \(n\)-term positive graphic sequence \(\pi = (d_1, d_2, \ldots, d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(H,n)\) has a realization \(G\) containing \(H\) as a subgraph. In particular, they pointed out that \(3n – 2 \leq \sigma(K_{4} – e, n) \leq 4n – 4\), where \(K_{r+1} – e\) denotes the graph obtained by removing one edge from the complete graph \(K_{r+1}\) on \(r+1\) vertices. Recently, Lai determined the values of \(\sigma(K_4 – e, n)\) for \(n \geq 4\). In this paper, we determine the values of \(\sigma(K_{r+1} – e, n)\) for \(r \geq 3\) and \(r+1 \leq n \leq 2r\), and give a lower bound of \(\sigma(K_{r+1} – e, n)\). In addition, we prove that \(\sigma(K_5 – e, n) = 5n – 6\) for even \(n\) and \(n \geq 10\) and \(\sigma(K_5 – e, n) = 5n – 7\) for odd \(n\) and \(n \geq 9\).
We show that if \(G\) is a \(3\)-connected graph of order at least \(5\), then there exists a longest cycle \(C\) of \(G\) such that the number of contractible edges of \(G\) which are on \(C\) is greater than or equal to \(\frac{|V(C)| + 9}{8}.\)
We obtain lower bounds for the number of elements dominated by a subgroup in a Cayley graph. Let \(G\) be a finite group and let \(U\) be a generating set for \(G\) such that \(U = U^{-1}\) and \(1 \in U\). Let \(A\) be an independent subgroup of \(G\). Let \(r\) be a positive integer, and suppose that, in the Cayley graph \((G,U)\), any two non-adjacent vertices have at most \(r\) common neighbours. Let \(N[H]\) denote the set of elements of \(G\) which are dominated by the elements of \(H\). We prove that
An interesting example illustrating these results is the graph on the symmetric group \(S_n\), in which two permutations are adjacent if one can be obtained from the other by moving one element. For this graph we show that \(r = 4\) and illustrate the inequalities.
Shapiro [8] asked what simple family of circuits will have resistances \(C_{2n}/{C_{2n}-1}\) (or something similar) where \(C_m=\frac{1}{m+1}\binom{2m}{m}\) is the \(m\)th Catalan number. In this paper, we give a construction of such circuits; we also discuss some related problems.
The two-dimensional bandwidth problem is to determine an embedding of graph \(G\) in a grid graph in the plane such that the longest edges are as short as possible. In this paper, we study the problem under the distance of \(L_\infty\)-norm.
Domination graphs of directed graphs have been defined and studied in a series of papers by Fisher, Lundgren, Guichard, Merz, and Reid. A tie in a tournament may be represented as a double arc in the tournament. In this paper, we examine domination graphs of tournaments, tournaments with double arcs, and more general digraphs.
In this paper, we consider the problem of decomposing complete multigraphs into multistars (a multistar is a star with multiple edges allowed). We obtain a criterion for the decomposition of the complete multigraph \(\lambda K_n\), into multistars with prescribed number of edges, but the multistars in the decomposition with the same number of edges are not necessarily isomorphic. We also consider the problem of decomposing \(\lambda K_n\) into isomorphic multistars and propose a conjecture about the decomposition of \(2K_n\) into isomorphic multistars.
For edges \(e\) and \(f\) in a connected graph \(G\), the distance \(d(e, f)\) between \(e\) and \(f\) is the minimum nonnegative integer \(n\) for which there exists a sequence \(e = e_0, e_1, \ldots, e_l = f\) of edges of \(G\) such that \(e_i\) and \(e_{i+1}\) are adjacent for \(i = 0, 1, \ldots, l-1\). Let \(c\) be a proper edge coloring of \(G\) using \(k\) distinct colors and let \(D = \{C_1, C_2, \ldots, C_k\}\) be an ordered partition of \(E(G)\) into the resulting edge color classes of \(c\). For an edge \(e\) of \(G\), the color code \(c_D(e)\) of \(e\) is the \(k\)-tuple \((d(e, C_1), d(e, C_2), \ldots, d(e, C_k))\), where \(d(e, C_i) = \min\{d(e, f): f \in C_i\}\) for \(1 \leq i \leq k\). If distinct edges have distinct color codes, then \(c\) is called a resolving edge coloring of \(G\). The resolving edge chromatic number \(\chi_{re}(G)\) is the minimum number of colors in a resolving edge coloring of \(G\). Bounds for the resolving edge chromatic number of a connected graph are established in terms of its size and diameter and in terms of its size and girth. All nontrivial connected graphs of size \(m\) with resolving edge chromatic number \(3\) or \(m\) are characterized. It is shown that for each pair \(k, m\) of integers with \(3 \leq k \leq m\), there exists a connected graph \(G\) of size \(m\) with \(\chi_{re}(G) = k\). Resolving edge chromatic numbers of complete graphs are studied.
A decomposition of optimal linear block codes with minimum distance \(d = 4\) and length \(4L\) into two subcodes is given such that one of the subcodes is an optimal length \(L\) code with minimum Hamming distance \(4\) and the other is a quasi-cyclic code of index \(4\). It is shown that the \(L\)-section minimal trellis diagram of the code is the product of the minimal trellis diagrams of the subcodes.
We consider the rank of the adjacency matrix of some classes of regular graphs that are transformed under certain unary operations. In particular, we study the ranks of the subdivision graph, the connected cycle graph, the connected subdivision graph, and the total graph of the following families of graphs: cycles, complete graphs, complete bipartite and multipartite graphs, circulant graphs of degrees three and four, and some Cartesian graph products.
For integers \( n \) and \( k \), we define \( r(n,k) \) as the average number of guesses needed to solve the game of Mastermind for \( n \) positions and \( k \) colours; and define \( f(n,k) \) as the maximum number of guesses needed. In this paper, we add more small values of the two parameters, and provide exact values for the case of \( n = 2 \). Finally, we comment on the asymptotics.
In this paper, we solve the existence problem for covering the \( 2 \)-paths of \( K_n \) with \( 4 \)-paths. This also settles the spectrum of \( 3 \)-path systems of the line graph of \( K_n \). The proof technique allows the embedding problem for \( (4, 2) \)-path coverings to be settled.
The general linear group \( G \) over \( \mathbb{Z}/2^n\mathbb{Z} \) acts transitively on the finite upper half plane over \( \mathbb{Z}/2^n\mathbb{Z} \), where \( \mathbb{Z} \) denotes the ring of rational integers. In this paper, it is shown that the pair of \( G \) and the stabilizer of a point on the plane is a Gelfand pair.
A \( c \)-partite tournament is an orientation of a complete \( c \)-partite graph. In 1991, Jian-zhong Wang conjectured that every arc of a regular 3-partite tournament \( D \) is contained in directed cycles of all lengths \( 3, 6, 9, \ldots, |V(D)| \).
In this paper, we show that this conjecture is completely false. Namely, for each integer \( t \) with \( 3 \leq t \leq |V(D)| \), we present an infinite family of regular 3-partite tournaments \( D \) such that there exists an arc in \( D \) which is not contained in a directed cycle of length \( t \).
Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A vertex labelling \( f: V \rightarrow \{0, 1, 2\} \) induces an edge labelling \( \overline{f}: E \rightarrow \{0, 1, 2\} \) defined by \( \overline{f}(uv) = |f(u) – f(v)| \). Let \( v_f(0), v_f(1), v_f(2) \) denote the number of vertices \( v \) with \( f(v) = 0, f(v) = 1 \) and \( f(v) = 2 \) respectively. Let \( e_f(0), e_f(1), e_f(2) \) be similarly defined. A graph is said to be 3-equitable if there exists a vertex labelling \( f \) such that \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for \( 0 \leq i, j \leq 2 \). In this paper, we show that every multiple shell \( MS\{n_1^{t_1}, \ldots, n_r^{t_r}\} \) is 3-equitable for all positive integers \( n_1, \ldots, n_r, t_1, \ldots, t_r \).
The rainbow Ramsey number \( RR(G_1, G_2) \) or constrained Ramsey number \( f(G_1,G_2) \) of two graphs \( G_1 \) and \( G_2 \) is defined to be the minimum integer \( N \) such that any edge-coloring of the complete graph \( K_N \) with any number of colors must contain either a subgraph isomorphic to \( G_1 \) with every edge the same color or a subgraph isomorphic to \( G_2 \) with every edge a different color. This number exists if and only if \( G_1 \) is a star or \( G_2 \) is acyclic. In this paper, we present the conjecture that the constrained Ramsey number of \( nK_2 \) and \( mK_2 \) is \( m(n-1)+2 \), along with a proof in the case \( m \leq \frac{3}{2}(n-1) \).
A set \( C \subseteq \mathbb{F}_2^n \) is said to be an asymmetric covering code with radius \( R \) if every word \( x \in \mathbb{F}_2^n \) can be obtained by replacing \( 1 \) by \( 0 \) in at most \( R \) coordinates of a word in \( C \). In this paper, tabu search is employed in the search for good asymmetric covering codes of small length. Fifteen new upper bounds on the minimum size of such codes are obtained in the range \( n \leq 13 \).
We use exhaustive computer searches to show that there are exactly \( 36 \) codewords in an optimal ternary \( (11,7) \) code and exactly \( 13 \) codewords in an optimal ternary \( (14,10) \) code. We also enumerate inequivalent optimal ternary \( (14,10) \) codes and show that there are exactly \( 6151 \) such codes.
The minimum number of blocks having maximum size precisely four that are required to cover, exactly \( \lambda \) times, all pairs of elements from a set of cardinality \( v \) is denoted by \( g_\lambda^{(4)}(v) \). We present a complete solution to this problem for \( v = 3, 4, \) and \( 5 \).
Among the well-studied maximal planar graphs, those having the maximum possible number of 3-cycles are precisely the planar chordal graphs (meaning no induced cycles of lengths greater than three). This motivates a somewhat similar result connecting maximal planar bipartite graphs, 4-cycles, and planar chordal bipartite graphs (meaning bipartite with no induced cycles of lengths greater than four), together with characterizations of planar chordal bipartite graphs as radial graphs of outerplanar multigraphs.
Combinatorial designs are a powerful tool because of their beautiful combinatorial structure that can help in many applications, such as coding theory or cryptography. A conference key distribution system is a scheme to design a conference key, and then to distribute this key to only participants attending the conference in order to communicate with each other securely. In this paper, we present an efficient conference key distribution system using difference families. Using techniques for creating the conference key and for performing authentication based on identification information, the communication protocol is designed. Applying the known results on difference families, we obtain many new infinite classes of conference key distribution systems. In special classes of difference families, the message overhead is \( O(v\sqrt{tv}) \), where \( v \) is the number of participants and \( t \) is the number of the \( k \)-elements subsets that consist of the difference family. The security of the presented protocol, which is an important problem in the construction of a secure system, is proved to be as computationally difficult to calculate as factoring and discrete logarithms.
In this paper, finite \( \{2,t\} \)-semiaffine linear spaces are investigated. When \( t = 5 \), their parameters are determined, and it is also proved that there is a single finite \( \{2, 5\} \)-semiaffine linear space on \( v = 20 \) points and with constant point degree \( 7 \).
We establish that for each of the 5005 possible types of 2-factorizations of the complete graph \( K_{13} \), there exists at least one solution. We also enumerate all nonisomorphic solutions to the Oberwolfach problem \( \text{OP}(13;3,3,3,4) \).
Scheduling static tasks on parallel architectures is a basic problem arising in the design of parallel algorithms. This NP-complete problem has been widely investigated in the literature and remains one of the most challenging questions in the field. Among the resolution methods for this type of problems, the taboo search technique is of particular interest. Based on this technique, two algorithms are proposed and tested on a sample of instances in order to be compared experimentally with other well-known algorithms. The results clearly indicate good overall performances of our algorithms. Next, some NP-completeness results are established showing that this problem is intractable for approximation, even for some restricted cases bearing a clear relation to the instances treated experimentally in this work.
\( X \)-proper edge colourings of bipartite graphs are defined. These colourings arise in timetables where rooms have to be assigned to courses. The objective is to minimize the number of different rooms in which each course must be taught. An optimum assignment is represented by a \( k \)-optimum edge colouring of a bipartite graph. Some necessary conditions for a \( k \)-optimum colouring are obtained, in terms of forbidden subgraphs. An algorithm based on removing these forbidden subgraphs to obtain improved colourings is described.
A graph \( G \) of order \( n \) is pancyclic if it contains a cycle of length \( \ell \) for every \( \ell \) such that \( 3 \leq \ell \leq n \). If the graph is bipartite, then it contains no cycles of odd length. A balanced bipartite graph \( G \) of order \( 2n \) is bipancyclic if it contains a cycle of length \( \ell \) for every even \( \ell \), such that \( 4 \leq \ell \leq 2n \). A graph \( G \) of order \( n \) is called \( k \)-semipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the cycle lengths in \( G \), i.e., for no \( \ell \leq n-k \) is it the case that each of \( C_\ell, \ldots, C_{\ell+k} \) is missing from \( G \). Generalizing this to bipartite graphs, a bipartite graph \( G \) of order \( n \) is called \( k \)-semibipancyclic, \( k \geq 0 \), if there is no “gap” of \( k+1 \) among the even cycle lengths in \( G \), i.e., for no \( \ell \leq n-2k \) is it the case that each of \( C_{2\ell}, \ldots, C_{2\ell+2k} \) is missing from \( G \).
In this paper we generalize a result of Hakimi and Schmeichel in several ways. First to \( k \)-semipancyclic, then to bipartite graphs, giving a condition for a hamiltonian bipartite graph to be bipancyclic or one of two exceptional graphs. Finally, we give a condition for a hamiltonian bipartite graph to be \( k \)-semibipancyclic or a member of a very special class of hamiltonian bipartite graphs.
We consider labeling edges of graphs with elements from abelian groups. Particular attention is given to graphs where the labels on any two Hamiltonian cycles sum to the same value. We find several characterizations for such labelings for cubes, complete graphs, and complete bipartite graphs. This extends work of \([1, 8, 9, 10]\). We also consider the computational complexity of testing if a labeled graph has this property and show it is NP-complete even when restricted to integer labelings of 3-connected, cubic, planar graphs with face girth at least five.
It is proved that the total chromatic number of any series-parallel graphs of degree at least \(3\) is \(\Delta(G)+1\).
We show that, in any coloring of the edges of \(K_{36}\), with two colors, there exists a triangle in the first color or a monochromatic \(K_{10}-e\) (\(K_{10}\) with one edge removed) in the second color, and hence we obtain a bound on the corresponding Ramsey number, \(R(K_3, K_{10}-e) \leq 38\). The new lower bound of \(37\) for this number is established by a coloring of \(K_{36}\) avoiding triangles in the first color and \(K_{10}-e\) in the second color. This improves by one the best previously known lower and upper bounds. We also give the bounds for the next Ramsey number of this type, \(42 \leq R(K_3, K_{11}-e) \leq 47\).
A subset \(S\) of \(V(G)\) is called a dominating set if every vertex in \(V(G) – S\) is adjacent to some vertex in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality taken over all dominating sets of \(G\). A dominating set \(S\) is called a tree dominating set if the induced subgraph \(\langle S\rangle\) is a tree. The tree domination number \(\gamma_{tr}(G)\) of \(G\) is the minimum cardinality taken over all minimal tree dominating sets of \(G\). In this paper, some exact values of tree domination number and some properties of tree domination are presented in Section [2]. Best possible bounds for the tree domination number, and graphs achieving these bounds are given in Section [3]. Relationships between the tree domination number and other domination invariants are explored in Section [4], and some open problems are given in Section [5].
If \(G\) is a tricyclic Hamiltonian graph of order \(n\) with maximum degree \(3\), then \(G\) has one of two forms, \(X(q,r,s,t)\) and \(Y(q,r,s,t)\), where \(q+r+s+t=n\). We find the graph \(G\) with maximal index by first identifying the graphs of each form having maximal index.
Let \(G = (V_1, V_2; E)\) be a bipartite graph with \(|V_1| = |V_2| = n \geq 2k\), where \(k\) is a positive integer. Let \(\sigma'(G) = \min\{d(u)+d(v): u\in V_1, v\in V_2, uv \not\in E(G)\}\). Suppose \(\sigma'(G) \geq 2k + 2\). In this paper, we will show that if \(n > 2k\), then \(G\) contains \(k\) independent cycles. If \(n = 2k\), then it contains \(k-1\) independent \(4\)-cycles and a \(4\)-path such that the path is independent of all the \(k-1\) \(4\)-cycles.
New results on the enumeration of noncrossing partitions with \(m\) fixed points are presented, using an enumeration polynomial \(P_m(x_1, x_2, \ldots, x_m)\). The double sequence of the coefficients \(a_{m,k}\) of each \(x^k_i\) in \(P_m\) is endowed with some important structural properties, which are used in order to determine the coefficient of each \(x^k_ix^l_j\) in \(P_m\).
This paper concerns a labeling problem of the plane graphs \(P_{a,b}\). We discuss the magic labeling of type \((1,1,1)\) and consecutive labeling of type \((1,1,1)\) of the graphs \(P_{a,b}\).
In this note, we prove that the largest non-contractible to \(K^p\) graph of order \(n\) with \(\lceil \frac{2n+3}{3} \rceil \leq p \leq n\) is the Turán’s graph \(T_{2p-n-1}(n)\). Furthermore, a new upper bound for this problem is determined.
If \(u\) and \(v\) are vertices of a graph, then \(d(u,v)\) denotes the distance from \(u\) to \(v\). Let \(S = \{v_1, v_2, \ldots, v_k\}\) be a set of vertices in a connected graph \(G\). For each \(v \in V(G)\), the \(k\)-vector \(c_S(v)\) is defined by \(c_S(v) = (d(v, v_1), d(v, v_2), \ldots, d(v, v_k))\). A dominating set \(S = \{v_1, v_2, \ldots, v_k\}\) in a connected graph \(G\) is a metric-locating-dominating set, or an MLD-set, if the \(k\)-vectors \(c_S(v)\) for \(v \in V(G)\) are distinct. The metric-location-domination number \(\gamma_M(G)\) of \(G\) is the minimum cardinality of an MLD-set in \(G\). We determine the metric-location-domination number of a tree in terms of its domination number. In particular, we show that \(\gamma(T) = \gamma_M(T)\) if and only if \(T\) contains no vertex that is adjacent to two or more end-vertices. We show that for a tree \(T\) the ratio \(\gamma_L(T)/\gamma_M(T)\) is bounded above by \(2\), where \(\gamma_L(G)\) is the location-domination number defined by Slater (Dominating and reference sets in graphs, J. Math. Phys. Sci. \(22 (1988), 445-455)\). We establish that if \(G\) is a connected graph of order \(n \geq 2\), then \(\gamma_M(G) = n-1\) if and only if \(G = K_{1,n-1}\) or \(G = K_n\). The connected graphs \(G\) of order \(n \geq 4\) for which \(\gamma_M(G) = n-2\) are characterized in terms of seven families of graphs.
The edges of a graph can be either directed or signed (\(2\)-colored) so as to make some of the even-length cycles of the underlying graph into alternating cycles. If a graph has a signing in which every even-length cycle is alternating, then it also has an orientation in which every even-length cycle is alternating, but not conversely. The existence of such an orientation or signing is closely related to the existence of an orientation in which every even-length cycle is a directed cycle.
We deal with the problem of labeling the vertices, edges, and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all \(s\)-sided faces constitute an arithmetic progression of difference \(d\). In this paper, we describe various antimagic labelings for the generalized Petersen graph \(P(n, 2)\). The paper concludes with a conjecture.
It was shown by Abrham that the number of pure Skolem sequences of order \(n\), \(n \equiv 0\) or \(1 \pmod{4}\), and the number of extended Skolem sequences of order \(n\), are both bounded below by \(2^{\left\lfloor \frac{n}{3} \right\rfloor}\). These results are extended to give similar lower bounds for the numbers of hooked Skolem sequences, split Skolem sequences, and split-hooked Skolem sequences.
Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph \(K_{a_1,a_2}\), with \(b_1\) roots in the first vertex class and \(b_2\) roots in the second vertex class. We give a simple proof of their formula, and a generalization for complete \(m\)-partite graphs, using the multivariate Lagrange inverse.
Using a linear space on \(v\) points with all block sizes \(|B| \equiv 0\) or \(1 \pmod{3}\), Doyen and Wilson construct a Steiner triple system on \(2v+1\) points that embeds a Steiner triple system on \(2|B|+1\) points for each block \(B\). We generalise this result to show that if the linear space on \(v\) points is extendable in a suitable way, there is a Steiner quadruple system on \(2v+2\) points that embeds a Steiner quadruple system on \(2(|B|+1)\) points for each block \(B\).
A graph with a graceful labeling (an \(\alpha\)-labeling) is called a graceful (\(\lambda\)-graceful) graph. In this paper, six methods for constructing bigger graceful graphs from a given graceful graph or a set of given \(\lambda\)-graceful graphs are provided. Two of which generalize Koh and others’ Theorems in [2, 3].
Let \(B_2\) be the bananas surface arising from the torus by contracting two different meridians of the torus to a simple point each. It was proved in [8] that there is not a finite Kuratowski theorem for \(B_2\).
A graph is outer-bananas-surface if it can be embedded in \(B_2\) so that all its vertices lie on the same face. In this paper, we prove that the class of the outer-\(B_2\) graphs is closed under minors. In fact, we give the complete set of \(38\) minor-minimal non-outer-\(B_2\) graphs and we also characterize these graphs by a finite list of forbidden topological minors.
We also extend outer embeddings to other pseudosurfaces. The \(S\) pseudosurfaces treated are spheres joined by points in such a way that each sphere has two singular points. We give an excluded minor characterization of outer-\(S\) graphs and we also give an explicit and finite list of forbidden topological minors for these pseudosurfaces.
We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig’s result on graphs with unique \(1\)-factors, a lemma by Seymour and Giles, theorems on alternating cycles in edge-colored graphs, and a theorem on semicycles in digraphs.
We consider computational problems related to the quoted results; all these problems ask whether a given (di)graph contains a cycle satisfying certain properties which runs through \(p\) prescribed vertices. We show that all considered problems can be solved in polynomial time for \(p < 2\) but are NP-complete for \(p \geq 2\).
We define a new graph operation called “dissolve \(N(v)\) into \(v\)” where \(N(v)\) is the set of vertices adjacent to a vertex \(v\) and characterize odd cycles of length greater than \(5\) in terms of \(p\)-critical graphs using this operation. This enables us to re-phrase the Strong Perfect Graph Conjecture,
Gray and Ramsay [5] showed that for any \(s \geq (2t – 1)2^t\), a \(t-(v,k)\) trade of volume \(s\) exists. In this note we improve their bound and show that for \(t \geq 3\), a given \(k\), and \(s \geq (t – 2)2^t + 2^{t-1} + 2\), there exists a simple \(t-(v,k)\) trade of volume \(s\).
\[S_{(p,x)} = \sum\limits_{k=0}^{n} {\binom{n}{k}}^p x^k\]
where \(n \geq 0\).
Then it is well-known that \(S_n(1,x), S_2(2,1), S_n(3,1)\) and \(S_n(3,1)\) can be exhibited in closed form. The formula
\[S_{2n}{(3,-1)} = (-1)^n\binom{2n}{n}\binom{3n}{n}\]
was discovered by A. C. Dixon in \(1891\). L. Carlitz [Mathematics Magazine, Vol. \(32 (1958), 47-48]\) posed the formulas
\[S_n{(3,1)}= ((x^n))(1-x^2)^nP_n(\frac{1+x}{1-x})\]
and
\[S_n{(4,1)} = ((x^n))(1-x)^{2n}\{P_n(\frac{1+x}{1-x})\}\]
where \(((x^n))f(x)\) means the coefficient of \(x^n\) in the series expansion of \(f(x)\). We use Legendre polynomials to get the analogous formulas
\[S_n{(3,-1)} = ((x^n))(1_x)^{2n}\]
and
\[S_n{(5,1)} = ((x^n))(1_x)^{2n}P_n(\frac{1+x}{1-x}S_n(3,x)\]
We obtain some partial results for \(S_n(p,x)\) when \(p\) is arbitrary, and also give a new proof of Dixon’s formula.
A graph \(H\) of order \(n\) is said to be embeddable in a graph \(G\) of order \(n\), if \(G\) contains a spanning subgraph isomorphic to \(H\). It is well known that any non-star tree \(T\) of order \(n\) is embeddable in its complement (i.e. in \(K_n – E(T)\)). In the paper “Packing two copies of a tree into its fourth power” by Hamamache Kheddouci, Jean-Francois Saclé, and Mariusz Wodgniak, Discrete Mathematics 213 (2000), 169-178, it is proved that any non-star tree \(T\) is embeddable in \(T^4 – E(T)\). They asked whether every non-star tree \(T\) is embeddable in \(T^3 – E(T)\). In this paper, answering their question negatively, we show that there exist trees \(T\) such that \(T\) is not embeddable in \(T^3 – E(T)\).
The linear \(2\)-arboricity \(la_2(G)\) of a graph \(G\) is the least integer \(k\) such that \(G\) can be partitioned into \(k\) edge-disjoint forests, whose component trees are paths of length at most \(2\). We prove that \(la_2(G) \leq \lfloor \frac{\Delta(G) + 4}{2} \rfloor\) if \(G\) is an outerplanar graph with maximum degree \(\Delta(G)\).
A paired-dominating set of a graph \(G\) is a dominating set of vertices whose induced subgraph has a perfect matching. We characterize the trees having unique minimum paired-dominating sets.
Given two graphs \(G\) and \(H \subseteq G\), we consider edge-colorings of \(G\) in which every copy of \(H\) has at least two edges of the same color. Let \(f(G,H)\) be the maximum number of colors used in such a coloring of \(E(G)\). Erdős, Simonovits, and Sós determined the asymptotic behavior of \(f\) when \(G = K_n\), and \(H\) contains no edge \(e\) with \(\chi(H – e) \leq 2\). We study the function \(f(G, H)\) when \(G = K_n\), or \(K_{m,n}\), and \(H\) is \(K_{2,t}\).
This article provides some new methods of construction of two and three associate class Nested Partially Balanced Incomplete Block (NPBIB) designs. The methods are based on Latin-square association scheme, rectangular association scheme, and triangular association scheme. One method of constructing NPBIB designs has also been given by incorporating a set of new treatments in place of each treatment in a Nested Balanced Incomplete Block (NBIB) design. Exhaustive catalogues of NPBIB designs based on two and three class association schemes with \(v \leq 30\) and \(r \leq 15\) have also been prepared.
A set \(D\) of vertices in a graph \(G\) is a total dominating set if every vertex of \(G\) has at least one neighbor in \(D\). The minimum cardinality of a total dominating set of \(G\) is called the total domination number of \(G\), denoted by \(\gamma_t(G)\). A total dominating set of \(G\) with cardinality \(\gamma_t(G)\) is called a \(\gamma_t\)-set of \(G\). We characterize trees with unique \(\gamma_t\)-sets. Further, we prove that \(\gamma_t(G) \leq \frac{3}{5}n(G)\) for graphs with unique \(\gamma_t\)-sets, and we characterize all graphs with unique \(\gamma_t\)-sets where \(\gamma_t(G) = \frac{3}{5}n(G)\).
A word \(w = w_1w_2\ldots w_n\) avoids an adjacent pattern \(\tau\) iff \(w\) has no subsequence of adjacent letters having all the same pairwise comparisons as \(\tau\). In [12] and [13] the concept of words and permutations avoiding a single adjacent pattern was introduced. We investigate the probability that words and permutations of length \(n\) avoid two or three adjacent patterns.
We consider a variant of what is known as the discrete isoperimetric problem, namely the problem of minimising the size of the boundary of a family of subsets of a finite set. We use the technique of `shifting’ to provide an alternative proof of a result of Hart. This technique was introduced in the early \(1980s\) by Frankl and Füredi and gave alternative proofs of previously known classical results like the discrete isoperimetric problem itself and the Kruskal-Katona theorem. Hence our purpose is to bring Hart’s result into this general framework.
The domatic number of a graph \(G\) is the maximum number of dominating sets into which the vertex set of \(G\) can be partitioned.
We show that the domatic number of a random \(r\)-regular graph is almost surely at most \(r\), and that for \(3\)-regular random graphs, the domatic number is almost surely equal to \(3\).
We also give a lower bound on the domatic number of a graph in terms of order, minimum degree, and maximum degree. As a corollary, we obtain the result that the domatic number of an \(r\)-regular graph is at least \((r+1)/(3ln(r+1))\).
The concept of circular chromatic number of graphs was introduced by Vince \((1988)\). In this paper, we define the circular chromatic number of uniform hypergraphs and study their basic properties. We study the relationship between the circular chromatic number, chromatic number, and fractional chromatic number of uniform hypergraphs.
For a given Hadamard design \(D\) of order \(n\), we construct another Hadamard design \(D’\) of the same order, which is disjoint from \(D\).
The existence question for the family of \(4-(15,5,\lambda)\) designs has long been answered for all values of \(\lambda\) except \(\lambda = 2\). Here, we resolve this last undecided case and prove that \(4-(15, 5, 2)\) designs are constructible.
In this note, we prove that a graph is of class one if \(G\) can be embedded in a surface with positive characteristic and satisfies one of the following conditions:(i) \(\Delta(G) \geq 3\) and \(g(G)\)(the girth of \(G\)) \(\geq 8\) (ii) \(\Delta(G) \geq 4\) and \(g(G) \geq 5\)(iii) \(\Delta(G) \geq 5\) and \(g(G) \geq 4\).
We investigate the optimization of a real-world logistics problem, which is concerned with shipping a dangerous chemical substance in various degrees of refinement to several locations and customers. Transport frequencies, inventories, and container flows have to be optimized. On the one hand, we discuss the mathematical structure of our problem (one result being its NP-completeness), and on the other hand, we describe our practical approach, which achieves nearly optimal solutions.
Let \( G \) be a \( k \)-regular graph of odd order \( n \geq 3 \) with \( k \geq \frac{n + 1}{2} \). This implies that \( k \) is even. Furthermore, let
\[
p = \min\left\{\frac{k}{2}, \left\lceil k-\frac{n}{3}\right\rceil\right\}.
\]
If \( x_1, x_2, \ldots, x_p \) are arbitrary given, pairwise different, vertices of the graph \( G \), then we show in this paper that there exist \( p \) pairwise edge-disjoint almost perfect matchings \( M_1, M_2, \ldots, M_p \) in \( G \) with the property that no edge of \( M_i \) is incident with \( x_i \) for \( i = 1, 2, \ldots, p \).
The previously studied notions of smart and foolproof finite order domination of a simple graph \( G = (V, E) \) are generalised in the sense that safe configurations in \( G \) are not merely sought after \( k \geq 1 \) moves, but in the limiting cases where \( k \to \infty \). Some general properties of these generalised domination parameters are established, after which the parameter values are found for certain simple graph structures (such as paths, cycles, multipartite graphs, and products of complete graphs, cycles, and paths).
Self-dual codes are an important class of linear codes. Hadamard matrices and weighing matrices have been used widely in the construction of binary and ternary self-dual codes. Recently, weighing matrices and orthogonal designs have been used to construct self-dual codes over larger fields. In this paper, we further investigate codes over \( \mathbb{F}_p \), constructed from orthogonal designs. Necessary conditions for these codes to be self-dual are established, and examples are given for lengths up to 40. Self-dual codes of lengths \( 2n \geq 16 \) over \( GF(31) \) and \( GF(37) \) are investigated here for the first time. We also show that codes obtained from orthogonal designs can generally give better results, with respect to their minimum Hamming distance, than codes obtained from Hadamard matrices, weighing matrices, or conference matrices.
We give decomposition formulas of the multiedge and the multipath zeta function of a regular covering of a graph \( G \) with respect to equivalence classes of prime, reduced cycles of \( G \). Furthermore, we give a decomposition formula of the weighted zeta function of a \( g \)-cyclic \( \Gamma \)-cover of a symmetric digraph \( D \) with respect to equivalence classes of prime cycles of \( D \), for any finite group \( \Gamma \) and \( g \in \Gamma \).
Let \( A \) be an abelian group. We call a graph \( G = (V, E) \) \( A \)-magic if there exists a labeling \( f : E(G) \to A^* \) such that the induced vertex set labeling \( f^+ : V(G) \to A \), defined by \( f^+(v) = \sum_{(u,v) \in E(G)} f(u,v) \), is a constant map. In this paper, we present some algebraic properties of \( A \)-magic graphs. Using them, various results are obtained for group-magic eulerian graphs.
Every Latin square of prime or prime power order \( s \) corresponds to a polynomial in 2 variables over the finite field on \( s \) elements, called the local permutation polynomial. What characterizes this polynomial is that its restrictions to one variable are permutations. We discuss the general form of local permutation polynomials and prove that their total degree is at most \( 2s – 4 \), and that this bound is sharp. We also show that the degree of the local permutation polynomial for Latin squares having a particular form is at most \( s – 2 \). This implies that circulant Latin squares of prime order \( p \) correspond to local permutation polynomials having degree at most \( p – 2 \). Finally, we discuss a special case of circulant Latin squares whose local permutation polynomial is linear in both variables.
Two graphs are said to be flow-equivalent if they have the same number of nowhere-zero \( \lambda \)-flows, i.e., they have the same flow polynomial. In this paper, we present a few methods of constructing non-isomorphic flow-equivalent graphs.
The Whitney number \( W_m{(n,k)} \) of the rank-\( n \) Dowling lattice \( Q_n(G) \) based on the group \( G \) having order \( m \) is the number of elements in \( Q_n(G) \) of co-rank \( k \). The associated numbers \( U_m{(n,k)} = k! W_m{(n,k)} \) and \( V_m{(n,k)} = k! m^k W_m{(n,k)} \) were studied by M. Benoumhani [\({Adv. in Appl. Math}\). 19 (1997), no. 1, 106-116] where a generating function was derived using algebraic techniques and logconcavity was shown for \( \{U_m{(n,k)}\} \) and for \( \{V_m{(n,k)}\} \). We give a central limit theorem and a local limit theorem on \( \mathbb{R} \) for \( \{U_m{(n,k)}\} \) and for \( \{V_m{(n,k)}\} \). In addition, asymptotic formulas for \( \max_k U_m{(n,k)} \), \( \max_k V_m{(n,k)} \) and their modes are given.
The Picard group is defined as \( \Gamma = SL(2, \mathbb{Z}[i]) \); the ring of \( 2 \times 2 \) matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients \( \Gamma/\Gamma(p) \) where \( p \) is a prime congruent to three mod four and \( \Gamma(p) \) is the congruence subgroup of level \( p \). We prove a decomposition theorem on the vertices of these graphs, and use this decomposition to derive upper and lower bounds on their isoperimetric numbers.
The domination number of a graph \( G \), \( \gamma(G) \), and the domination graph of a digraph \( D \), \( dom(D) \), are integrated in this paper. The \( \gamma \)-set di domination graph of the complete biorientation of a graph \( G \), \( dom_{\gamma}(\overset{\leftrightarrow}{G}) \), is created. All \( \gamma \)-sets of specific trees \( T \) are found, and \( dom_{\gamma}(\overset{\leftrightarrow}{T}) \) is characterized for those classes.
A fractional automorphism of a graph is a doubly stochastic matrix which commutes with the adjacency matrix of the graph. If we apply an ordinary automorphism to a set of vertices with a particular property, such as being independent or dominating, the resulting set retains that property. We examine the circumstances under which fractional automorphisms preserve the fractional properties of functions on the vertex set.
A king graph \( KG_n \) has \( n^2 \) vertices corresponding to the \( n^2 \) squares of an \( n \times n \) chessboard. From one square (vertex) there are edges to all squares (vertices) being attacked by a king. For given graphs \( G \) and \( H \), the Ramsey number \( r(G, H) \) is the smallest \( n \) such that any 2-coloring of the edges of \( KG_n \) contains \( G \) in the first or \( H \) in the second color. Results on existence and nonexistence of \( r(G, H) \) and some exact values are presented.
A set \( \{a_1,a_2,\ldots,a_n\} \) of positive integers with \( a_1 < a_2 < \cdots < a_n \) is said to be equi-graphical if there exists a graph with exactly \( a_i \) vertices of degree \( a_i \) for each \( i \) with \( 1 \leq i \leq n \). It is known that such a set is equi-graphical if and only if \( \sum_{i=1}^{n} a_i \) is even and \( a_n \leq \sum_{i=1}^{n-1} a_i^2 \). This concept is generalized to the following problem: Given a set \( S \) of positive integers and a permutation \( \pi \) on \( S \), determine when there exists a graph containing exactly \( a_i \) vertices of degree \( \pi(a_i) \) for each \( i \) (\( 1 \leq i \leq n \)). If such a graph exists, then \( \pi \) is called a graphical permutation. In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation \( \pi \) of \( S = \{a_1,a_2,\ldots,a_n\} \), where \( 1 \leq a_1 < a_2 < \cdots < a_n \) and such that \( \pi(a_n) = a_n \), is graphical if and only if \( \sum_{i=1}^{n} a_i\pi(a_i) \) is even and \( a_n \leq \sum_{i=1}^{n-1} a_i\pi(a_i) \).
The formula for the number of spanning trees in \( K_{t_1,\ldots,t_P} \) is well known. In this paper, we give an algorithm that generates the list of spanning trees in \( K_{s,t} \).
For any \( k \in \mathbb{N} \), a graph \( G = (V,E) \) is said to be \( \mathbb{Z}_k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k – \{0\} \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \) defined by
\[
l^+(v) = \sum_{u \in N(v)} l(uv)
\]
is a constant map. For a given graph \( G \), the set of all \( k \in \mathbb{Z}_+ \) for which \( G \) is \( \mathbb{Z}_k \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will consider trees whose diameters are at most \( 4 \) and will determine their integer-magic spectra.
In this paper, by using the generating function method, we obtain a series of identities involving the generalized Fibonacci and Lucas numbers.
This paper introduces the problem of finding a permutation \(\phi\) on the vertex set \(V(G)\) of a graph \(G\) such that the sum of the distances from each vertex to its image under \(\phi\) is maximized. We let \(\mathcal{S}(G) = \max \sum_{v\in V(G)} d(v, \phi(v))\), where the maximum is taken over all permutations \(\phi\) of \(V(G)\). Explicit formulae for several classes of graphs as well as general bounds are presented.
The local-edge-connectivity \((u,v)\) of two vertices \(u\) and \(v\) in a graph or digraph \(D\) is the maximum number of edge-disjoint \(u-v\) paths in \(D\), and the edge-connectivity of \(D\) is defined as \(\lambda(D) = \min\{\lambda(u, v) | u,v \in V(D)\}\). Clearly, \(\lambda(u,v) \leq \min\{d^+(u),d^-(v)\}\) for all pairs \(u\) and \(v\) of vertices in \(D\). We call a graph or digraph \(D\) maximally local-edge-connected when
\[\lambda(u, v) = \min\{d^+(u),d^-(v)\}\]
for all pairs \(u\) and \(v\) of vertices in \(D\).
Recently, Fricke, Oellermann, and Swart have shown that some known sufficient conditions that guarantee equality of \(\lambda(G)\) and minimum degree \(\delta(G)\) for a graph \(G\) are also sufficient to guarantee that \(G\) is maximally local-edge-connected.
In this paper we extend some results of Fricke, Oellermann, and Swart to digraphs and we present further sufficient conditions for
graphs and digraphs to be maximally local-edge-connected.
We show that every hamiltonian claw-free graph with a vertex \(x\) of degree \(d(x) \geq 7\) has a \(2\)-factor consisting of exactly two cycles.
This paper presents two new algorithms for generating \((n,2)\) de Bruijn sequences which possess certain properties. The sequences generated by the proposed algorithms may be useful for experimenters to systematically investigate intertrial repetition effects. Characteristics are compared with those of randomly sampled \((n,2)\) de Bruijn sequences.
Let \(\alpha(G)\) and \(\tau(G)\) denote the independence number and matching number of a graph \(G\), respectively. The tensor product of graphs \(G\) and \(H\) is denoted by \(G \times H\). Let \(\underline{\alpha}(G \times H) = \max \{\alpha(G) \cdot n(H), \alpha(H) \cdot n(G)\}\) and \(\underline{\tau}(G \times H) = 2\tau(G) \cdot \tau(H)\), where \(\nu(G)\) denotes the number of vertices of \(G\). It is easy to see that \(\alpha(G \times H) \geq \underline{\alpha}(G \times H)\) and \(\beta(G \times H) \geq \underline{\tau}(G \times H)\). Several sufficient conditions for \(\alpha(G \times H) > \underline{\alpha}(G \times H)\) are established. Further, a characterization is established for \(\alpha(G \times H) = \underline{\tau}(G \times H)\). We have also obtained a necessary condition for \(\alpha(G \times H) = \underline{\alpha}(G \times H)\). Moreover, it is shown that neither the hamiltonicity of both \(G\) and \(H\) nor large connectivity of both \(G\) and \(H\) can guarantee the equality of \(\alpha(G \times H)\) and \(\underline{\alpha}(G \times H)\).
A graphoidal cover of a graph \(G\) is a collection \(\psi\) of (not necessarily open) paths in \(G\) such that every vertex of \(G\) is an internal vertex of at most one path in \(\psi\) and every edge of \(G\) is an exactly one path in \(\psi\). If further no member of \(\psi\) is a cycle, then \(\psi\) is called an acyclic graphoidal cover of \(G\). The minimum cardinality of an acyclic graphoidal cover is called the acyclic graphoidal covering number of \(G\) and is denoted by \(\eta_a\). In this paper, we characterize the class of graphs for which \(\eta_a = q – p\), where \(p\) and \(q\) denote respectively the order and size of \(G\).
The dissociation number of a graph \(G\) is the number of vertices in a maximum size induced subgraph of \(G\) with vertex degrees at most \(1\). The problem of finding the dissociation number was introduced by Yannakakis, who proved it is NP-hard on the class of bipartite graphs. In this paper, we analyze the dissociation number problem restricted to the class of bipartite graphs in more detail. We strengthen the result of Yannakakis by reducing the problem, in polynomial time, from general bipartite graphs to some particular classes, such as bipartite graphs with maximum degree \(3\) or \(C_4\)-free bipartite graphs. Besides the negative results, we prove that finding the dissociation number is polynomially solvable for bipartite graphs containing no induced subgraph isomorphic to a tree with exactly three vertices of degree \(1\) of distances \(1\), \(2\), and \(3\) from the only vertex of degree \(3\).
The induced matching number of a graph \(G\) is the number of edges in a maximum size induced subgraph of \(G\) with vertex degrees equal to \(1\). Analogous results hold for the induced matching number.
A vertex \(k\)-coloring of a graph \(G\) is acyclic if no cycle is bichromatic. The minimum integer \(k\) such that \(G\) admits an acyclic \(k\)-coloring is called the acyclic chromatic number of \(G\), denoted by \(\chi_a(G)\). In this paper, we discuss some properties of maximal acyclic \(k\)-colorable graphs, prove a sharp lower bound of the \(\chi_a(G)\) and get some results about the relation between \(\chi(G)\) and \(\chi_a(G)\). Furthermore, a conjecture of B. Grünbaum that \(\chi_a(G) \leq \Delta+1\) is proved for maximal acyclic \(k\)-colorable graphs.
In this paper, we focus on the existence of \(2\)-critical sets in the latin square corresponding to the elementary abelian \(2\)-group of order \(2^n\). It has been shown by Stinson and van Rees that this latin square contains a \(2\)-critical set of volume \(4^n – 3^n\). We provide constructions for \(2\)-critical sets containing \(4^n – 3^n + 1 – \left(2^{k-1} + 2^{m-1} + 2^{n-(k+m+1)}\right)\) entries, where \(1 \leq k \leq n\) and \(1 \leq m \leq n – k\). That is, we construct \(2\)-critical sets for certain values less than \(4^n – 3^n + 1 – 3\cdot 2^{\lfloor n/3\rfloor – 1}\). The results raise the interesting question of whether, for the given latin square, it is possible to construct \(2\)-critical sets of volume \(m\), where \(4^n – 3^n + 1 – 3\cdot 2^{\lfloor n/3\rfloor – 1} < m < 4^n – 3^n\).
In this paper, we find explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets \(S_{n}(T; \tau)\) of all permutations in \(S_n\) that contain \(\tau \in S_k\) exactly once and avoid a subset \(T \subseteq S_3\) where \(|T| \geq 2\).
A digraph \(T\) is called strongly connected if for every pair of vertices \(u\) and \(v\) there exists a directed path from \(u\) to \(v\) and a directed path from \(v\) to \(u\). Denote the in-degree and out-degree of a vertex \(v\) of \(T\) by \(d^-(v)\) and \(d^+(v)\), respectively. We define \(\delta^- = \min_{v\in V(T)} \{d^-(v)\}\), and \(\delta_+ = \min_{v\in V(T)} \{d^+(v)\}\). Let \(T_0\) be a \(7\)-tournament which contains no transitive \(4\)-subtournament. Let \(T\) be a strong tournament, \(T \ncong T_0\) and \(k \geq 2\). In this paper, we show that if \(\delta^+ + \delta^- \geq \frac{k-2}{k-1}n+3k(k-1)\), then \(T\) can be partitioned into \(k\) cycles. When \(n \geq 3k(k-1)\) a regular strong \(n\)-tournament can be partitioned into \(k\) cycles and a almost regular strong \(n\)-tournament can be partitioned into \(k\) cycles when \(n \geq (3k+1)(k-1)\). Finally, if a strong tournament \(T\) can be partitioned into \(k\) cycles, \(q\) is an arbitrary positive integer not larger than \(k\). We prove that \(T\) can be partitioned into \(q\) cycles.
Let \(G = (V, E)\) be a simple graph. Let \(\alpha\) and \(\mathrm{IR}\) be the independence number and upper irredundance number of \(G\), respectively. In this paper, we prove that for any graph \(G\) of order \(n\) with maximum degree \(\Delta \geq 1\), \(\mathrm{IR}(G) – \alpha(G) \leq \frac{\Delta -2}{2\Delta }n\). When \(\Delta = 3\), the result was conjectured by Rautenbach.
We first establish the relationship between the largest eigenvalue of the Laplacian matrix of a graph and its bipartite density. Then, we present lower and upper bounds for the largest Laplacian eigenvalue of a graph in terms of its largest degree and diameter.
In this paper, we prove the gracefulness of the class of graphs denoted by \(\mathcal{P}_{a,b}\).
Let \(D\) be a dominating set of a simple graph \(G = (V, E)\). If the subgraph \((V – D)_G\)induced by the set \(V – D\) is disconnected, then \(D\) is called a split dominating set of \(G\), and if \(\langle D\rangle_G\) has no edges, then \(D\) is an independent dominating set of \(G\). If every vertex in \(V\) is adjacent to some vertex of \(D\) in \(G\), then \(D\) is a total dominating set of \(G\). The split domination number \(\gamma_s(G)\), independent domination number \(i(G)\), and total domination number \(\gamma_t(G)\) equal the minimum cardinalities of a split, independent, and total dominating set of \(G\), respectively. The concept of split domination was first defined by Kulli and Janakiram in 1997 [4], while total domination was introduced by Cockayne, Dawes, and Hedetniemi in 1980 [2].
In this paper, we study the split, independent, and total domination numbers of corona \(G \circ H\) and generalized coronas \(kG \circ H\) of graphs.
A set of points in a Steiner triple system (STS(\(v\))) is said to be independent if no three of these points occur in the same block. In this paper, we derive for each \(k \leq 8\) a closed formula for the number of independent sets of cardinality \(k\) in an STS(\(v\)). We use the formula to prove that every STS(21) has an independent set of cardinality eight and is, as a consequence, \(4\)-colourable.
Let \(G\) be a graph with \(n\) vertices and let \(D\) be a minimum dominating set of \(G\). If \(V – D\) contains a dominating set \(D’\) of \(G\), then \(D’\) is called an inverse dominating set of \(G\) with respect to \(D\). The inverse domination number \(\gamma'(G)\) of \(G\) is the cardinality of a smallest inverse dominating set of \(G\). In this paper, we characterise graphs for which \(\gamma(G) + \gamma'(G) = n\). We give a lower bound for the inverse domination number of a tree and give a constructive characterisation of those trees which achieve this lower bound.
Siri and Gvozdjak proved in [9] that the bananas surface, the pseudosurface consisting in the \(2\)-amalgamation of two spheres, does not admit a finite Kuratowski Theorem.
In this paper we prove that pseudosurfaces arising from the \(n\)-amalgamation of two closed surfaces, \(n \geq 2\), do not admit a finite Kuratowski Theorem, by showing an infinite family of minimal non-embeddable graphs.
We denote by \(K(l*r)\) the complete \(r\)-partite graph with \(l\) vertices in each part, and denote \(K(l*v)+K(m*s)+K(n*t)+\cdots\) by \(K(l*r,m*s,n*t,\ldots)\). Kierstead showed that the choice number of \(K(3*r)\) is exactly \(\left\lceil\frac{4r-1}{3}\right\rceil\). In this paper, we shall determine the choice number of \(K(3*r,1*t)\), and consider the choice number of \(K(3*r,2*s,1*t)\).
For any prime power \(q\), there exists an affine plane of order \(q\). The complement of an affine plane is a balanced incomplete block design (BIBD) with block size \(q^2-q\). In this note, a proof is given that the blocks can be split into sub-blocks to form a nested BIBD with parameters \((q^2, q^2+q, q^3+q^2, q^2-1,q-1)\). Alternatively, this is a generalized tournament design with one game each round, involving \(q\) teams, each team with \(q-1\) players.
Let \(\mathcal{F}\) be a family of \(k\)-graphs. A \(k\)-graph \(G\) is called \(\mathcal{F}\)-saturated if it is a maximal graph not containing any member of \(\mathcal{F}\) as a subgraph. We investigate the smallest number of edges that an \(\mathcal{F}\)-saturated graph on \(n\) vertices can have. We present new results and open problems for different instances of \(\mathcal{F}\).
A strongly regular vertex with parameters \((\lambda, \mu)\) in a graph is a vertex \(x\) such that the number of neighbors any other vertex \(y\) has in common with \(x\) is \(\lambda\) if \(y\) is adjacent to \(x\), and is \(\mu\) if \(y\) is not adjacent to \(x\). In this note, we will prove some basic properties of these vertices and the graphs that contain them, as well as provide some simple constructions of regular graphs that are not necessarily strongly regular, but do contain many strongly regular vertices. We also make several conjectures and find all regular graphs on at most ten vertices with at least one strongly regular vertex.
Given \(S\) a benzenoid system, we find an expression of the second order Randić index, denoted by \(\mathop{^2\chi}(S)\), in terms of inlet features of \(S\). As a consequence, we classify benzenoid systems with equal \(\mathop{^2\chi}\) and then find the minimal and maximal value over the set of catacondensed systems.
Let \(m_1, m_2, \ldots, m_r\) be positive integers with \(m_i \geq 3\) for all \(i\). An \((m_1, m_2, \ldots, m_r)\)-cycle is defined as the edge-disjoint union of \(r\) cycles of lengths \(2m_1, 2m_2, \ldots, 2m_r\). An \((m_1, m_2, \ldots, m_r)\)-cycle system of the complete graph \(K_n\) is a decomposition of \(K_n\) into \((m_1, m_2, \ldots, m_r)\)-cycles.
In this paper, the necessary and sufficient conditions for the existence of an \((m_1, m_2, \ldots, m_r)\)-cycle system of \(K_n\) are given, where \(m_i\) \((1 \leq i \leq r)\) are odd integers with \(3 \leq m_i \leq n\) and \(\sum_{i=1}^r m_i = 2^k\) for \(k \geq 3\). Moreover, the complete graph with a \(1\)-factor removed \(K_n – F\) has a similar result.
We refer to a labeling of a plane graph as a d-antimagic labeling if the vertices, edges and faces of the graph are labeled in such a way that the label of a face and the labels of vertices and edges surrounding that face add up to a weight of the face and the weights
of faces constitute an arithmetical progression of difference \(d\). In this paper we deal with \(d\)-antimagic labeling of prisms.
We show that the classical Ramsey number \(R(3,3,3,3)\) is no greater than \(62\). That is, any edge coloring with four colors of a complete graph on \(62\) vertices must contain a monochromatic triangle. Basic notions and a historical overview are given along with the theoretical framework underlying the main result. The algorithms for the computational verification of the result are presented along with a brief discussion of the software tools that were utilized.
We show that certain subsets of \(\mathbf{F}_q\)-rational points of the curve \(XZ^{n-1} = Y^n\) are dense sets in \(\mathbf{P}^2(\mathbf{F}_q)\).
H. Kharaghani, in “Arrays for orthogonal designs” \((2000)\), \(\textit{J. Combin. Designs}\), \(8 (2000), 166-173\), showed how to use amicable sets of matrices to construct orthogonal designs in orders divisible by eight. We show how amicable orthogonal designs can be used to make amicable sets and so obtain infinite families of orthogonal designs in six variables in orders divisible by eight.
Let \(G\) and \(H\) be a pair of non-isomorphic graphs on fewer than \(m\) vertices. In this paper, we introduce several new problems about decomposing the complete graph \(K_m\) into copies of \(G\) and \(H\). We will assume that at least one of \(G\) or \(H\) is not a cycle. We also begin to examine variations to the problems of subgraph packing, covering, and factorization.
For vertices \(u\) and \(v\) in a connected graph \(G\) with vertex set \(V\), the distance \(d(u,v)\) is the length of a shortest \(u – v\) path in \(G\). A \(u – v\) path of length \(d(u,v)\) is called a \(u – v\) geodesic. The closed interval \(I[u,v]\) consists of \(u\), \(v\), and all vertices that lie in some \(u – v\) geodesic of \(G\); while for \(S \subseteq V\), \(I[S]\) is the union of closed intervals \(I[u,v]\) for all \(u,v \in S\). A set \(S\) of vertices is a geodetic set if \(I[S] = V\), and the minimum cardinality of a geodetic set is the geodetic number \(g(G)\). For vertices \(x\) and \(y\) in \(G\), the detour distance \(D(x, y)\) is the length of a longest \(x – y\) path in \(G\). An \(x – y\) path of length \(D(x, y)\) is called an \(x – y\) detour. The closed detour interval \(I_D[x,y]\) consists of \(x\), \(y\), and all vertices in some \(x – y\) detour of \(G\). For \(S \subseteq V\), \(I_D[S]\) is the union of \(I_D[x,y]\) for all \(x,y \in S\). A set \(S\) of vertices is a detour set if \(I_D[S] = V\), and the minimum cardinality of a detour set is the detour number \(dn(G)\). We study relationships that can exist between minimum detour sets and minimum geodetic sets in a graph. A graph \(F\) is a minimum detour subgraph if there exists a graph \(G\) containing \(F\) as an induced subgraph such that \(V(F)\) is a minimum detour set in \(G\). It is shown that \(K_3\) and \(P_3\) are minimum detour subgraphs. It is also shown that for every pair \(a,b \geq 2\) of integers, there exists a connected graph \(G\) with \(dn(G) = a\) and \(g(G) = b\).
We use a dynamic programming algorithm to establish a lower bound on the domination number of complete grid graphs \( G_{m,n} \). The bound is within \( 5 \) of a known upper bound that has been conjectured to be the exact domination number of the complete grid graphs.
A large set of KTS(\(v\)), denoted by LKTS(\(v\)), is a collection of (\(v-2\)) pairwise disjoint KTS(\(v\)) on the same set. In this paper, it is proved that there exists an LKTS(\(3^n \cdot 91\)) for any integer \(n \geq 1\).
If \( x \) is a vertex of a digraph \( D \), then we denote by \( d^+(x) \) and \( d^-(x) \) the outdegree and the indegree of \( x \), respectively. The global irregularity of a digraph \( D \) is defined by \(i_g(D) = \max\{d^+(x), d^-(x)\} – \min\{d^+(y), d^-(y)\}\) over all vertices \( x \) and \( y \) of \( D \) (including \( x = y \)). If \( i_g(D) = 0 \), then \( D \) is regular, and if \( i_g(D) \leq 1 \), then \( D \) is called almost regular. The local irregularity is defined as \(i_l(D) = \max[|d^+(x) – d^-(x)|]\) over all vertices \( x \) of \( D \). The path covering number of \( D \) is the minimum number of directed paths in \( D \) that are pairwise vertex disjoint and cover the vertices of \( D \). A semicomplete \( c \)-partite digraph is a digraph obtained from a complete \( c \)-partite graph by replacing each edge with an arc, or a pair of mutually opposite arcs with the same end vertices. If a semicomplete \( c \)-partite digraph \( D \) does not contain an oriented cycle of length two, then \( D \) is called a \( c \)-partite tournament.
In 2000, Gutin and Yeo [7] proved sufficient conditions for the local irregularity of a semicomplete multipartite digraph to secure a path covering number of at most \( k \). In this paper, we will give a useful supplement to this result by using bounds for the global irregularity that guarantee a path covering number of at most \( k \). As an application, we will present sufficient conditions for close to regular multipartite tournaments containing a Hamiltonian path. Especially, we will characterize almost regular \( c \)-partite tournaments containing a Hamiltonian path.
An extended \(7\)-cycle system of order \( n \) is an ordered pair \( (V, B) \), where \( B \) is a collection of edge-disjoint 7-cycles, 3-tadpoles, and loops which partition the edges of the graph \( K_n^+ \) whose vertex set is an \( n \)-set \( V \). In this paper, we show that an extended 7-cycle system of order \( n \) exists for all \( n \) except \( n = 2, 3, \) and \( 5 \).
A grid graph is a finite induced subgraph of the infinite 2-dimensional grid defined by \( \mathbb{Z} \times \mathbb{Z} \) and all edges between pairs of vertices from \( \mathbb{Z} \times \mathbb{Z} \) at Euclidean distance precisely \( 1 \). An \( m \times n \)-rectangular grid graph is induced by all vertices with coordinates from \( 1 \) to \( m \) and from \( 1 \) to \( n \), respectively. A natural drawing of a (rectangular) grid graph \( G \) is obtained by drawing its vertices in \( \mathbb{R}^2 \) according to their coordinates. We consider a subclass of the rectangular grid graphs obtained by deleting some vertices from the corners. Apart from the outer face, all (inner) faces of these graphs have area one (bounded by a \( 4 \)-cycle) in a natural drawing of these graphs. We determine which of these graphs contain a Hamilton cycle, i.e., a cycle containing all vertices, and solve the problem of determining a spanning \( 2 \)-connected subgraph with as few edges as possible for all these graphs.
The (previously studied) notions of secure domination and of weak Roman domination involve the construction of protection strategies in a simple graph \( G = (V, E) \), by utilizing the minimum number of guards needed at vertices in \( V \) to protect \( G \) in different scenarios (these minimum numbers are called the secure [weak Roman] domination parameters for the graph). In this paper, these notions are generalized in the sense that safe configurations in \( G \) are not merely sought after one move, but rather after each of \( k \geq 1 \) moves. Some general properties of these generalized domination parameters are established, after which the parameter values are found for certain simple graph structures (such as paths, cycles, multipartite graphs, and products of complete graphs, cycles, and paths).
We establish necessary and sufficient conditions on \( m \) and \( n \) for \( K_m \times K_n \), the Cartesian product of two complete graphs, to be decomposable into cycles of length \( 8 \). We also provide a complete classification of the leaves that are possible with maximum packings of complete graphs with \( 8 \)-cycles.
We consider the rank of the adjacency matrix of the line graph for some classes of regular graphs. In particular, we study the line graphs of cycles, paths, complete graphs, complete bipartite and multipartite graphs, circulant graphs of degrees three and four, and some Cartesian graph products.
For each vertex \( v \) in a graph \( G \), let there be associated a particular type of a subgraph \( F_v \) of \( G \). In this context, the vertex \( v \) is said to dominate \( F_v \). A set \( S \) of vertices of \( G \) is called a full dominating set if every vertex of \( G \) belongs to a subgraph \( F_v \) of \( G \) for some \( v \in S \) and every edge of \( G \) belongs to a subgraph \( F_w \) of \( G \) for some \( w \in S \). The minimum cardinality of a full dominating set of \( G \) is its full domination number \( \gamma_F(G) \). A full dominating set of \( G \) of cardinality \( \gamma_F(G) \) is called a \( \gamma_F \)-set of \( G \).
We study three types of full domination in graphs: full star domination, where \( F_v \) is the maximum star centered at \( v \); full closed domination, where \( F_v \) is the subgraph induced by the closed neighborhood of \( v \); and full open domination, where \( F_v \) is the subgraph induced by the open neighborhood of \( v \).
A subset \( T \) of a \( \gamma_F \)-set \( S \) in a graph \( G \) is a forcing subset for \( S \) if \( S \) is the unique \( \gamma_F \)-set containing \( T \). The forcing full domination number of \( S \) in \( G \) is the minimum cardinality of a forcing subset for \( S \), and the forcing full domination number \( f_{\gamma_F}(G) \) of the graph \( G \) is the minimum forcing full domination number among all \( \gamma_F \)-sets of \( G \).
We present several realization results concerning forcing parameters in full domination.
A minimum feedback arc set of a digraph is a smallest sized set of arcs whose reversal makes the resulting digraph acyclic. Given an acyclic digraph \( D \), we seek a smallest sized tournament \( T \) having \( A(D) \) as a minimum feedback arc set. The reversing number of a digraph \( D \) equals \( |V(T)| – |V(D)| \). We investigate the reversing number of the \( k \)th power of directed Hamiltonian path \( P_n^k \), when \( k \) is fixed and \( n \) tends to infinity. We show that even for small values of \( k \), where \( |A(P_n^k)| \) is much closer to \( |A(P_n)| \) than \( |A(T_n)| \), the opposite relationship holds for the reversing number.
A union closed (UC) family \( \mathcal{A} \) is a finite family of sets such that the union of any two sets in \( \mathcal{A} \) is also in \( \mathcal{A} \). Peter Frankl conjectured in 1979 that for every union closed family \( \mathcal{A} \), there exists some \( x \) contained in at least half the members of \( \mathcal{A} \). In this paper, we show that if a UC family \( \mathcal{A} \) fails the conjecture, then no element can appear in more than two of its \( 3 \)-sets, and so the number of \( 3 \)-sets in \( \mathcal{A} \) can be no more than \( \frac{2n}{3} \).
A \( (k,d) \)-total coloring (\( k,d \in \mathbb{N}, k \geq 2d \)) of a graph \( G \) is an assignment \( c \) of colors \( \{0,1,\ldots,k-1\} \) to the vertices and edges of \( G \) such that \( d \leq |c(x_i) – c(x_j)| \leq k – d \) whenever \( x_i \) and \( x_j \) are two adjacent edges, two adjacent vertices, or an edge incident to a vertex. The circular total chromatic number \( \chi_c”(G) \) is defined by \(\chi_c”(G) = \inf\{k/d : G \text{ has a } (k, d)\text{-total coloring}\}.\) It was proved that \( \chi”(G) – 1 < \chi_c''(G) \leq \chi''(G) \) — where \( \chi''(G) \) is the total chromatic number of \( G \) — with equality for all type-1 graphs and most of the so far considered type-2 graphs. We determine an infinite class of graphs \( G \) such that \( \chi_c''(G) < \chi''(G) \) and we list all graphs of order \( <7 \) with this property.
In this paper we consider a variation of the classical Turán-type extremal problems. Let \( S \) be an \( n \)-term graphical sequence, and \( \sigma(S) \) be the sum of the terms in \( S \). Let \( H \) be a graph. The problem is to determine the smallest even \( l \) such that any \( n \)-term graphical sequence \( S \) having \( \sigma(S) \geq l \) has a realization containing \( H \) as a subgraph. Denote this value \( l \) by \( \sigma(H, n) \). We show \(\sigma(C_{2m+1}, n) = m(2n – m – 1) + 2, \quad \text{for } m \geq 3, n \geq 3m;\) \(\sigma(C_{2m+2}, n) = m(2n – m – 1) + 4, \quad \text{for } m \geq 3, n \geq 5m – 2. \)
We first prove that if \( G \) is a connected graph with \( n \) vertices and chromatic number \( \chi(G) = k \geq 2 \), then its independent domination number
\[i(G) \leq \left\lceil \frac{(k-1)}{k}n \right\rceil – (k-2).\]
This bound is tight and remains so for planar graphs. We then prove that the independent domination number of a diameter two planar graph on \( n \) vertices is at most \( \left\lceil \frac{n}{3} \right\rceil \).
Hill, Landjev, Jones, Storme, and Barat proved in a previous article on caps in \(PG(5, 3)\) and \(PG(6,3)\) that every 53-cap in \(PG(5, 3)\) is contained in the 56-cap of Hill and that there exist complete 48-caps in \(PG(5,3)\). The first result was used to lower the upper bound on \( m_2(6,3) \) on the size of caps in \(PG(6, 3)\) from 164 to 154. Presently, the known upper bound on \( m_2(6, 3) \) is 148. In this article, using computer searches, we prove that every 49-cap in \(PG(5, 3)\) is contained in a 56-cap, and that every 48-cap, having a 20-hyperplane with at most 8-solids, is also contained in a 56-cap. Computer searches for caps in \(PG(6,3)\) which use the computer results of \(PG(5,3)\) then lower the upper bound on \( m_2(6,3) \) to \( m_2(6,3) \leq 136 \). So now we know that \( 112 \leq m_2(6,3) \leq 136 \).
Let \( \delta(G) \) and \( \lambda(G) \) be the minimum degree and edge-connectivity of a graph \( G \), respectively. A graph \( G \) is maximally edge-connected if \( \lambda(G) = \delta(G) \) and super-edge-connected if every minimum edge cut consists of edges adjacent to a vertex of minimum degree.
In this paper, sufficient conditions for super-edge-connected graphs depending on the clique number and the minimum degree are presented. These results show that some known sufficient conditions for maximally edge-connected graphs even lead to super-edge-connected graphs.
For a vertex \(v\) of a connected graph \(G\) and a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v, S) = \min\{d(v,x) : x \in S\}\), where \(d(v,x)\) is the distance between \(v\) and \(x\). For an ordered \(k\)-partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) of \(V(G)\), the code of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(c_\Pi(v) = (d(v,S_1), d(v,S_2), \ldots, d(v, S_k))\). The \(k\)-partition \(\Pi\) is a resolving partition if the codes \(c_\Pi(v)\), \(v \in V(G)\), are distinct. A resolving partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) is acyclic if each subgraph \(\langle S_i \rangle\) induced by \(S_i\) (\(1 \leq i \leq k\)) is acyclic in \(G\). The minimum \(k\) for which there is a resolving acyclic \(k\)-partition of \(V(G)\) is the resolving acyclic number \(a_r(G)\) of \(G\). We study connected graphs with prescribed order, diameter, vertex-arboricity, and resolving acyclic number. It is shown that, for each triple \(d,k,n\) of integers with \(2 \leq d \leq n-2\) and \(3 \leq (n-d+1)/2 \leq k \leq n-d+1\), there exists a connected graph of order \(n\) having diameter \(d\) and resolving acyclic number \(k\). Also, for each pair \(a, b\) of integers with \(2 \leq a \leq b-1\), there exists a connected graph with resolving acyclic number \(a\) and vertex-arboricity \(b\). We present a sharp lower bound for the resolving acyclic number of a connected graph in terms of its clique number. The resolving acyclic number of the Cartesian product \(H \times K_2\) of nontrivial connected graph \(H\) and \(K_2\) is studied.
In this paper, we completely solve the problem of finding a maximum packing of any balanced complete multipartite graph \(K_{m}(n)\) with edge-disjoint \(6\)-cycles, and minimum leaves are explicitly given.
Subsequently, we also find a minimum covering of \(K_{m}(n)\).
Orthogonal designs and their special cases, such as weighing matrices and Hadamard matrices, have many applications in combinatorics, statistics, and coding theory, as well as in signal processing. In this paper, we generalize the definition of orthogonal designs, give many constructions for these designs, and prove some multiplication theorems that, most of them, can also be applied in the special case of orthogonal designs. Some necessary conditions for the existence of generalized orthogonal designs are also given.
We prove that the corona graphs \(C_n \circ K_1\) are \(k\)-equitable, as per Cahit’s definition of \(k\)-equitability, for \(k = 2, 3, 4, 5, 6\).
For a vertex \(v\) of a graph \(G = (V, E)\), the domination number \(\gamma(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a dominating set in \(G\) that contains \(v\). The average domination number of \(G\) is \(\gamma_{av}(G) = \frac{1}{|V|} \sum_{v\in V} \gamma_v(G)\). The independent domination number \(i_v(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a maximal independent set in \(G\) that contains \(v\). The average independent domination number of \(G\) is \(\gamma_{av}^i(G) = \frac{1}{|V|} \sum_{v\in V} i_v(G)\). In this paper, we show that a tree \(T\) satisfies \(\gamma_{av}(T) = i_{av}(T)\) if and only if \(A(T) = \vartheta\) or each vertex of \(A(T)\) has degree \(2\) in \(T\), where \(A(T)\) is the set of vertices of \(T\) that are contained in all its minimum dominating sets.
A graph \(G\) is \(K_r\)-covered if each vertex of \(G\) is contained in a clique \(K_r\). Let \(\gamma(G)\) and \(\gamma_t(G)\) respectively denote the domination and the total domination number of \(G\). We prove the following results for any graph \(G\) of order \(n\):
If \(G\) is \(K_6\)-covered, then \(\gamma_t(G) \leq \frac{n}{3}\),
If \(G\) is \(K_r\)-covered with \(r = 3\) or \(4\) and has no component isomorphic to \(K_r\), then \(\gamma_t(G) \leq \frac{2n}{r+1}\),
If \(G\) is \(K_3\)-covered and has no component isomorphic to \(K_3\), then \(\gamma(G) + \gamma_t(G) \leq \frac{7n}{9}\).
Corollaries of the last two results are that every claw-free graph of order \(n\) and minimum degree at least \(3\) satisfies \(\gamma_t(G) \leq \frac{n}{2}\) and \(\gamma(G) + \gamma(G) \leq \frac{7n}{9}\). For general values of \(r\), we give conjectures which would generalise the previous results. They are inspired by conjectures of Henning and Swart related to less classical parameters \(\gamma_{K_r}\) and \(\gamma^t_{K_r}\).
We are interested in linear-fractional transformations \(y,t\) satisfying the relations \(y^6=t^6 = 1\), with a view to studying an action of the subgroup \(H = \) on \({Q}(\sqrt{n}) \cup \{\infty\}\) by using coset diagrams.
For a fixed non-square positive integer \(n\), if an element \(\alpha = \frac{a+\sqrt {n}}{c}\) and its algebraic conjugate have different signs, then \(\alpha\) is called an ambiguous number. They play an important role in the study of action of the group \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\). In the action of \(H\) on \({Q}(\sqrt{n}) \cup \{\infty\}\), \(\mathrm{Stab}_\alpha{(H)}\) are the only non-trivial stabilizers and in the orbit \(\alpha H\); there is only one (up to isomorphism). We classify all the ambiguous numbers in the orbit and use this information to see whether the action is transitive or not.
We are studying clique graphs of planar graphs, \(K(\text{Planar})\), this means the graphs which are the intersection of the clique family of some planar graph. In this paper, we characterize the \(K_3\) – free and \(K_4\) – free graphs which are in \(K(\text{Planar})\).
We show that a self-complementary vertex-transitive graph of order \(pq\), where \(p\) and \(q\) are distinct primes, is isomorphic to a circulant graph of order \(pq\). We will also show that if \(\Gamma\) is a self-complementary Cayley graph of the nonabelian group \(G\) of order \(pq\), then \(\Gamma\) and the complement of \(\Gamma\) are not isomorphic by a group automorphism of \(G\).
One of the most important problems of coding theory is to construct codes with the best possible minimum distance. The class of quasi-cyclic codes has proved to be a good source for such codes. In this paper, we use the algebraic structure of quasi-cyclic codes and the BCH type bound introduced in [17] to search for quasi-cyclic codes which improve the minimum distances of the best-known linear codes. We construct \(11\) new linear codes over \(\text{GF}(8)\) where \(3\) of these codes are one unit away from being optimal.
A graph \(G\) is said to be \(locally\) \(hamiltonian\) if the subgraph induced by the neighbourhood of every vertex is hamiltonian. Alabdullatif conjectured that every connected locally hamiltonian graph contains a spanning plane triangulation. We disprove the conjecture. At the end, we raise a problem about the nonexistence of spanning planar triangulation in a class of graphs.
Recently, Babson and Steingrimsson (see \([BS]\)) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.
In this paper we study the generating functions for the number of permutations on \(n\) letters avoiding a generalized pattern \(ab-c\) where \((a,b,c) \in S_3\), and containing a prescribed number of occurrences of a generalized pattern \(cd-e\) where \((c,d,e) \in S_3\). As a consequence, we derive all the previously known results for this kind of problem, as well as many new results.
Let \(G = (V,E)\) be a simple graph. For any real valued function \(f:V \to {R}\) and \(S \subset V\), let \(f(S) = \sum_{v\in S} f(u)\). A signed \(k\)-subdominating function is a function \(f: V \to \{-1,1\}\) such that \(f(N[v]) \geq 1\) for at least \(k\) vertices \(v \in V\). The signed \(k\)-subdomination number of a graph \(G\) is \(\gamma_{ks}^{-11}(G) = \min \{f(V) | f \text{ is a signed } k\text{-subdominating function on } G\}\). In this paper, we obtain lower bounds on this parameter and extend some results in other papers.
We give some relationships among the intersection numbers of a distance-regular graph \(\Gamma\) which contains a circuit \((u_1,u_2,u_3,u_4)\) with \(\partial(u_1,u_2) = 1\) and \(\partial(u_2,u_4) = 2\). As an application, we obtain an upper bound of the diameter of \(\Gamma\) when \(k \geq 2b_1\).
We extend results concerning orthogonal edge labeling of constant weight Gray codes. For positive integers \(n\) and \(r\) with \(n > r\), let \(G_{n,r}\) be the graph whose vertices are the \(r\)-sets of \(\{1, \ldots, n\}\), with \(r\)-sets adjacent if they intersect in \(r-1\) elements. The graph \(G_{n,r}\) is Hamiltonian; Hamiltonian cycles of \(G_{n,r}\) are early examples of error-correcting codes, where they came to be known as constant weight Gray codes.
An \(r\)-set \(A\) and a partition \(\pi\) of weight \(r\) are said to be orthogonal if every block of \(\pi\) meets \(A\) in exactly one element. Given a class \(P\) of weight \(r\) partitions of \(X_n\), one would like to know if there exists a \(G_{n,r}\) Hamiltonian cycle \(A_1 A_2 \ldots A_{\binom{n}{r}}\) whose edges admit a labeling \(A_1\pi_1 A_2 \ldots A_{\binom{n}{r}}\pi_{\binom{n}{r}}\) by distinct partitions from \(\mathcal{P}\), such that a partition label of an edge is orthogonal to the vertices that comprise the edge. The answer provides non-trivial information about Hamiltonian cycles in \(G_{n,r}\) and has application to questions pertaining to the efficient generation of finite semigroups.
Let \(r\) be a partition of \(m\) as a sum of \(r\) positive integers. We let \(r\) also refer to the set of all partitions of \(X_n\) whose block sizes comprise the partition \(r\). J. Lehel and the first author have conjectured that for \(n \geq 6\) and partition type \(\pi\) of \(\{1, \ldots, n\}\) of weight \(r\) partitions, there exists a \(r\)-labeled Hamiltonian cycle in \(G_{n,r}\).
In the present paper, for \(n = s + r\), we prove that there exist Hamiltonian cycles in \(G_{n,r}\) which admit orthogonal labelings by the partition types which have \(s\) blocks of size two and \(r – s\) blocks of size one, thereby extending a result of J. Lehel and the first author and completing the work on the conjecture for all partition types with blocks of size at most two.
For distinct vertices \(u\) and \(v\) of a nontrivial connected graph \(G\), the detour distance \(D(u,v)\) between \(u\) and \(v\) is the length of a longest \(u-v\) path in \(G\). For a vertex \(v \in V(G)\), define \(D^-(v) = \min\{D(u,v) : u \in V(G) – \{v\}\}\). A vertex \(u (\neq v)\) is called a detour neighbor of \(v\) if \(D(u,v) = D^-(v)\). A vertex \(v\) is said to detour dominate a vertex \(u\) if \(u = v\) or \(u\) is a detour neighbor of \(v\). A set \(S\) of vertices of \(G\) is called a detour dominating set if every vertex of \(G\) is detour dominated by some vertex in \(S\). A detour dominating set of \(G\) of minimum cardinality is a minimum detour dominating set and this cardinality is the detour domination number \(\gamma_D(G)\). We show that if \(G\) is a connected graph of order \(n \geq 3\), then \(\gamma_D(G) \leq n-2\). Moreover, for every pair \(k,n\) of integers with \(1 \leq k \leq n-2\), there exists a connected graph \(G\) of order \(n\) such that \(\gamma_D(G) = k\). It is also shown that for each pair \(a,b\) of positive integers, there is a connected graph \(G\) with domination number \(\gamma(G) = a\) and \(\gamma_D(G) = b\).
We let \(A(n)\) equal the number of \(n \times n\) alternating sign matrices. From the work of a variety of sources, we know that
\[A(n) = \prod\limits_{t=0}^{n-1} \frac{(3l+1)!}{(n+l)!}\]
We find an efficient method of determining \(ord_p(A(n))\), the highest power of \(p\) which divides \(A(n)\), for a given prime \(p\) and positive integer \(n\), which allows us to efficiently compute the prime factorization of \(A(n)\). We then use our method to show that for any nonnegative integer \(k\), and for any prime \(p > 3\), there are infinitely many positive integers \(n\) such that \(ord_p(A(n)) = k\). We show a similar but weaker theorem for the prime \(p = 3\), and note that the opposite is true for \(p = 2\).
We survey the status of minimal coverings of pairs with block sizes two, three, and four when \(\lambda = 1\), that is, all pairs from a \(v\)-set are covered exactly once. Then we provide a complete solution for the case \(\lambda = 2\).
We derive an alternative rule for generating uniform step magic squares. The compatibility conditions for the proposed rule are simpler than the analogous conditions for the classical uniform step rule. We exploit this fact to enumerate all uniform-step magic squares of every given odd order. Our main result states that if \(p = \prod_{i=1}^l q_i^{r_i}\) is the prime factorization of a positive odd number \(p\), then there exist \(\kappa(p) =\prod _{i=1}^l \kappa(q_i^{r_i})\) uniform step magic squares of order \(p\), where
\(\kappa (q_i^{r_i})=[\tau (q_i^{r_i})]^2-\lambda (q_i^{r_i}),\lambda(q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})^2[2(q_i^{2r_i-1}+1)^2/(q_i+1)^2+q_i^{3r_i-1}(q_i^{r_i}-3q_i^{r_i-1})]\) and \(\tau (q_i^{r_i})=(q_i^{r_i}-q_i^{r_i-1})(q_i^{2r_i+1}-2q_i^{2r_i}-q_i^{2r_i-1}+2)/(q_i+1)\) for \(i=1,\ldots,l\)
We show that for a cubic graph on \(n\) nodes, the size of the dominating set found by the greedy algorithm is at most \(\frac{4}{9}n\), and that this bound is tight.
For a standard tableau \(T\) of shape \(\lambda \vdash n\), \(maj(T)\) is the sum of \(i\)’s such that \(i+1\) appears in a row strictly below that of \(i\) in \(T\). We consider the \(g\)-polynomial \(f^\lambda(q) = \sum_\tau q^{ maj(T)}\), which appears in many contexts: as a dimension of an irreducible representation of finite general linear group, as a special case of Kostka-Foulkes polynomials, and so on. In this article, we try to understand `maj’ on a standard tableau \(T\) in relation to `inv’ on a multiset permutation (or a permutation of type \(\lambda\)). We construct an injective map from the set of standard tableaux to the set of permutations of type \(\lambda\) (increasing in each block) such that the `maj’ of the tableau is the `maj’ of the corresponding permutation when \(\lambda\) is a two-part partition. We believe that this helps to understand irreducible unipotent representations of finite general linear groups.
Many results about outer-embeddings (graphs having all their vertices in the same face) have been obtained recently in topological graph theory in recent times. In this paper, we deal with some difficulties appearing in the study of such embeddings. Particularly, we propose several problems concerning outer-embeddings in pseudosurfaces and we prove that two of them are NP-complete.
We also describe some properties about lists of forbidden minors for outer-embeddings in certain kinds of pseudosurfaces.
For some fixed \( n_0 \geq 0 \), we study the minimum number of vertices or edges that have to be removed from a graph such that no component of the rest has more than \( n_0 \) vertices.
An \( [r, s, n, t] \)-configuration is a collection \(C\) of \(r\)-sets in \( \{1, \ldots, n\} \) such that every \( s \)-set in \( \{1, \ldots, n\} \) contains at most \( t \) of the \( r \)-sets in \( C \). Studying this generalization of the Steiner system was suggested by a theorem of Poonen on union-closed families of sets. In this paper, we consider only \( [3, 4, n, 2] \)-configurations, and refer to them as \(n\)-configurations; by an \( (n, k) \)-configuration we mean an \(n\)-configuration containing exactly \(k\) \(3\)-sets. An \((n,k)\)-configuration is maximal if it is not contained in any \( (n, k + 1) \)-configuration; finally, \( L(n) \) is the largest integer \(k\) for which an \((n, k)\)-configuration exists. In this paper, we determine \(L(n)\) for \( 4 \leq n \leq 9 \), and characterize all the maximal \( n \)-configurations for \(n = 4, 5,\) and \(6\), as well as the \((n, L(n))\)-configurations for \( n = 7, 8, \) and \( 9 \).
Let \( G \) be a simple graph with vertex set \( V \) and edge set \( E \). A vertex labeling \( f: V \to \{0,1,2\} \) induces an edge labeling \( \bar{f}: E \to \{0,1,2\} \) defined by \( \bar{f}(uv) = |f(u) – f(v)| \). Let \( u_f(i) \) denote the number of vertices \( v \) with \( f(v) = i \), \( i = 0,1,2 \). Similarly, \( e_f(i) \) denotes the number of edges \( uv \) with \( \bar{f}(uv) = i \), \( i = 0,1,2 \). A graph is said to be \( 3 \)-equitable if there exists a vertex labeling \( f \) such that \( |v_f(i) – v_f(j)| \leq 1 \) and \( |e_f(i) – e_f(j)| \leq 1 \) for all \( i \neq j \), \( i, j = 0,1,2 \). In which case, \( f \) is called a \( 3 \)-equitable labeling.
In this paper, we prove that the following graphs are three equitable: (1) Helm graph \( H_n \) (\( n \geq 4 \)), (2) A Flower graph \( FL_n \), (3) One point union \( H_n^{(k)} \) of \( k \)-copies of \( H_n \), \( k \geq 1 \), (4) One point union \( K_4^{(k)} \) of \( k \) copies of \( K_4 \), (5) A \( K_4 \)-snake of \( n \) blocks, each equal to \( K_4 \), (6) A \( C_t \)-snake of \( n \) blocks, \( t = 4,6 \) and \( t = 5 \) with \( n \) not congruent to \( 3 \) modulo \( 6 \).
A defensive alliance in a graph \( G = (V,E) \) is a set of vertices \( S \subseteq V \) satisfying the condition that every vertex \( v \in S \) has at most one more neighbor in \( V – S \) than it has in \( S \). Because of such an alliance, the vertices in \( S \), agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in \( V – S \). In this paper, we introduce this new concept, together with a variety of other kinds of alliances, and initiate the study of their mathematical properties.
The distance-\( k \) domination number of graph \( G \), \( \gamma_{\leq k}(G) \), is the cardinality of a smallest set of vertices, \( S \), such that every vertex not in \( S \) is no more than distance \( k \) from at least one vertex of \( S \). Carrington, Harary, and Haynes showed \( |V^0| \geq 2|V^+| \) where \( V^0 = \{u \in V: \gamma_{\leq 1}(G-v) = \gamma_{\leq 1}(G)\} \) and \( V^+ = \{v \in V: \gamma_{\leq 1}(G-v) > \gamma_{\leq 1}(G)\} \). This paper extends the result to distance-\( k \) domination, with the obvious change in definition of \( V^0 \) and \( V^+ \), to show \( |V^0| \geq \frac{2}{2k-1}|V^+| \). Extremal graphs are characterized when \( k = 1 \) and some progress is mentioned on the characterization problem when \( k > 1 \).
We investigate constraints in finite Boolean lattices \( \langle \mathcal{P}(X), \subseteq \rangle \) where \( X \) is a finite set. The constraints studied here are of the form \( \langle Z,k \rangle \) where \( Z \subseteq X \), \( 1 \leq k \leq |Z| \). A set \( I \subseteq X \) \({satisfies}\) \( \langle Z,k \rangle \) if \( |I \cap Z| \geq k \). We characterize the sets satisfying collections of such constraints as filters (final segments) in \( \langle \mathcal{P}(X) \rangle \). We find yet other characterizations of filters including one by means of families of sets indexed by elements of \( X \) so that the elements of the filter correspond to subfamilies with an empty intersection. Our characterizations are supported for algorithms. We also study the families of negated constraints and mixed families and find their characterizations. In the positive case, formulas built of constraints can be used to measure the complexity of filters (and thus also of antichains of their minimal elements). We find pathological filters with very simple descriptions when the disjunctions are allowed, but extremely complex descriptions when only conjunctions are allowed.
The number \( g_3^{(4)}(v) \) represents the minimum cardinality of a pairwise balanced design on \( v \) elements in which the largest block size is four and every pair occurs exactly three times. We give a survey of the results for this quantity.
Let \( G_1 \) and \( G_2 \) be any two 2-regular graphs, each with \( n \) vertices. Let \( G \) be any cubic graph obtained from \( G_1 \) and \( G_2 \) by adding \( n \) edges, each of which joins a vertex in \( G_1 \) to a vertex in \( G_2 \). We show that \( G \) has a myriad of vertex-magic total labelings, with at least three different magic constants. This class of cubic graphs includes all generalized Petersen graphs.
Let [n, k, d]q codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). In this paper, the existence of the following codes is proven: [42, 6, 30]8, [49, 6, 36]8, [78, 6, 60]8, [84, 6, 65]8, [91, 6, 71]8, [96, 6, 75]8, [102, 6, 80]8, [108, 6, 85]8, [114, 6, 90]8,and [48, 6, 35]9, [54, 6, 40]9, [60, 6, 45]9, [96, 6, 75]9, [102, 6, 81]9, [108, 6, 85]9, [114, 6, 90]9, [126, 6, 100]9, [132, 6, 105]9. The nonexistence of five codes over GF(9) is also proven. All of these results improve the respective upper and lower bounds in Brouwer’s table [2].
In this paper, we obtain some necessary existence conditions for bi-level balanced arrays of strength six by using some classical inequalities and by expressing the moments of the weights of the columns of such arrays in terms of its parameters. We present some illustrative examples to compare these results with the earlier known results.
This paper describes a comprehensive approach to the analysis and synthesis of tree-structured communication networks. First, a class of models for tree-structured communication networks is proposed. Then, performance parameters such as communication delays and network reliability are defined, and efficient algorithms for calculating these parameters are provided. Subsequently, an application of a powerful tree-generating algorithm to the synthesis of optimal communication networks is described. The universal approach of this algorithm allows for its use in conjunction with the proposed model and the algorithms for calculating values of performance parameters. The paper shows sample optimal tree-structured networks resulting from applying the synthesis algorithm for various optimization parameters.
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 \( G \). The subgraph of \( G \) induced by its eccentric vertices is the eccentric subgraph 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 neighbor \( w \) of \( v \). A vertex \( v \) is a boundary vertex of \( G \) if \( v \) is a boundary vertex of some vertex of \( G \). The subgraph of \( G \) induced by its boundary vertices is the boundary of \( G \). A vertex \( v \) is an interior vertex of \( G \) if for every vertex \( u \) distinct from \( v \), there exists a vertex \( w \) distinct from \( v \) such that \( d(u,w) = d(u,v) + d(v,w) \). The interior of \( G \) is the subgraph of \( G \) induced by its interior vertices. A vertex \( v \) is a boundary vertex of a connected graph if and only if \( v \) is not an interior vertex. For every graph \( G \), there exists a connected graph \( H \) such that \( G \) is both the center and interior of \( H \).
Relationships between the boundary and the periphery, center, and eccentric subgraph of a graph are studied. The boundary degree of a vertex \( v \) in a connected graph \( G \) is the number of vertices \( u \) in \( G \) having \( v \) as a boundary vertex. We study, for each pair \( r,n \) of integers with \( r \geq 0 \) and \( n \geq 3 \), the existence of a connected graph \( G \) of order \( n \) such that every vertex of \( G \) has boundary degree \( r \). We also study the boundary vertices of a connected graph from different points of view.
Let \( G \) be a simple graph having a maximum matching \( M \). The deficiency \( \text{def}(G) \) of \( G \) is the number of vertices unsaturated by \( M \). A bridge in a connected graph \( G \) is an edge \( e \) of \( G \) such that \( G-e \) is disconnected. A graph is said to be almost cubic (or almost 3-regular) if one of its vertices has degree \( 3 + e \), \( e \geq 0 \), and the others have degree 3. In this paper, we find the minimum number of bridges of connected almost cubic graphs with a given deficiency.
The cardinality of the minimal pairwise balanced designs on \( v \) elements with largest block size \( k \) is denoted by \( g^{(k)}(v) \). It is known that \(30 \leq g^{(4)}(18) \leq 33.\)In this note, we show that \(31 \leq g^{(4)}(18).\)
In this paper, we introduce two new classes of critical sets, \( t \)-uniform and \( T \)-uniform (where \( t \) is a positive integer and \( T \) is a partial Latin square). We identify, up to isomorphism, all \( t \)-uniform critical sets of order \( n \), where \( 2 \leq n \leq 6 \). We show that the completable product of two \( T \)-uniform critical sets is a \( T \)-uniform critical set for certain partial Latin squares \( T \), and then apply this theorem to small examples to generate infinite families of \( T \)-uniform critical sets.
Let \(G\) be a graph with integral edge weights. A function \(d: V(G) \to \mathbb{Z}_p\) is called a nowhere \(0 \mod p\) domination function if each \(v \in V\) satisfies \((\sum_{u \in N(v)} w(u,v)d(u))\neq 0 \mod p\), where \(w(u,v)\) denotes the weight of the edge \((u,v)\) and \(N(v)\) is the neighborhood of \(v\). The subset of vertices with \(d(v) \neq 0\) is called a nowhere \(0 \mod p\) dominating set. It is known that every graph has a nowhere \(0 \mod 2\) dominating set. It is known to be false for all other primes \(p\). The problem is open for all odd \(p\) in case all weights are one.
In this paper, we prove that every unicyclic graph (a graph containing at most one cycle) has a nowhere \(0 \mod p\) dominating set for all \(p > 1\). In fact, for trees and cycles with any integral edge weights, or for any other unicyclic graph with no edge weight of \((-1) \mod p\), there is a nowhere \(0 \mod p\) domination function \(d\) taking only \(0-1\) values. This is the first nontrivial infinite family of graphs for which this property is established. We also determine the minimal graphs for which there does not exist a \(0 \mod p\) dominating set for all \(p > 1\) in both the general case and the \(0-1\) case.
We apply the technique of patchwork embeddings to find orientable genus embeddings of the Cartesian product of a complete regular tripartite graph with an even cycle. In particular, the orientable genus of \(K_{m,m,m} \times C_{2n}\) is determined for \(m \geq 1\) and for all \(n \geq 3\) and \(n = 1 \). For \(n = 2\) both lower and upper bounds are given.
We see that the resulting embeddings may have a mixture of triangular and quadrilateral faces, in contrast to previous applications of the patchwork method.
The redundancy \(R(G)\) of a graph \(G\) is the minimum, over all dominating sets \(S\), of \(\sum_{v \in S} 1 + d(v)\), where \(d(v)\) is the degree of vertex \(v\). We establish a sharp upper bound on the redundancy of trees and characterize all trees that achieve the bound.
We first prove that for any fixed \(k\), a cubic graph with few short cycles contains a \(K_{k}\)-minor. This is a direct generalization of a result on girth by Thomassen. We then use this theorem to show that for any fixed \(k\), a random cubic graph contains a \(K_{k}\)-minor asymptotically almost surely.
Partial parallelisrms Uhat admit a collineation group that fixes one spread \(\Sigma\), fixes a line of it and acts sharply two-transitive on the remaining lines of \(\Sigma\) are completely classified.
Recently, Babson and Steingrimsson (see \([BS]\)) introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation.
Following \([BCS]\), let \(e_k,m\) (respectively, \(f_k\pi\)) be the number of occurrences of the generalized pattern \(12-3-\ldots-k\) (respectively, \(21-3-\ldots-k\)) in a permutation \(\pi\). In the present note, we study the distribution of the statistics \(e_k,f_k\) and \(f_k\pi\) in a permutation avoiding the classical pattern \(1-3-2\).
We also present some applications of our results, which relate the enumeration of permutations avoiding the classical pattern \(1-3-2\) according to the statistics \(e_k\) and \(f_k\) to Narayana numbers and Catalan numbers.
We show that a negation of tautology corresponds to a family of graphs without nowhere-zero group- and integer-valued flows.
We show that in any graph \(G\) on \(n\) vertices with \(d(x) + d(y) \geq n\) for any two nonadjacent vertices \(x\) and \(y\), we can fix the order of \(k\) vertices on a given cycle and find a Hamiltonian cycle encountering these vertices in the same order, as long as \(k < n/12\) and \(G\) is \([(k+1)/2]\)-connected. Further, we show that every \([3k/2]\)-connected graph on \(n\) vertices with \(d(x) + d(y) \geq n\) for any two nonadjacent vertices \(x\) and \(y\) is \(k\)-ordered Hamiltonian, i.e., for every ordered set of \(k\) vertices, we can find a Hamiltonian cycle encountering these vertices in the given order. Both connectivity bounds are best possible.
We establish that for any \(m \in \mathbb{N}\) and any \(K_m\)-free graph \(G\) on \(\mathbb{N}\), there exist large additive and multiplicative structures that are independent with respect to \(G\). In particular, there exists for each \(l \in \mathbb{N}\) an arithmetic progression \(A_l\) of length \(l\) with increment chosen from the finite sums of a prespecified sequence \(\langle t_{l,n}\rangle _{n=1}^{\infty}\), such that \(\bigcup_{i=1 }^\infty A_l\) is an independent set. Moreover, if \(F\) and \(H\) are disjoint finite subsets of \(\mathbb{N}\), and for each \(t \in F \cup H\), \(a_t \in A_l\), then \(\{\Sigma_{t \in F}a_t\Sigma_{t \in H} a_t\}\) is not an edge of \(G\). If \(G\) is \(K_{m,m}\)-free, one may drop the disjointness assumption on the sets \(F\) and \(H\). Analogous results are valid for geometric progressions.
A connected graph \(G(V, E)\) is said to be \((a, d)\)-antimagic if there exist positive integers \(a\) and \(d\) and a bijection \(f: E \to \{1, 2, \ldots, |E|\}\) such that the induced mapping \(g_f: V \to \mathbb{N}\) defined by \(g_f(v) = \sum\{f(u,v) | (u, v) \in E(G)\}\) is injective and \(g_f(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\}\). In this paper, we mainly investigate \((a, d)\)-antimagic labeling of some special trees, complete bipartite graphs \(K_{m,n}\), and categorize \((a, d)\)-antimagic unicyclic graphs.
A graph \(G = (V, E)\) is said to be an \(integral \;sum \;graph\) ( respectively, \(sum \;graph\)) if there is a labeling \(f\) of its vertices with distinct integers ( respectively, positive integers) , so that for any two vertices \(u\) and \(v\), \(uv\) is an edge of \(G\) if and only if \(f(u) + f(v) = f(w)\) for some other vertex \(w\). For a given graph \(G\), the \(integral\; sum\; number\) \(\zeta = \zeta(G)\) (respectively, \(sum\; number\) \(\sigma = \sigma(G)\) ) is defined to be the smallest number of isolated vertices which when added to \(G\) result in an integral sum graph (respectively, sum graph). In a graph \(G\), a vertex \(v \in V(G)\) is said to a \(hanging\; vertex\) if the degree of it \(d(v) = 1\). A path \(P \subseteq G\), \(P = x_ox_1x_2\ldots x_t\), is said to be a \(hanging\; path\) if its two end vertices are respectively a hanging vertex \(x_o\) and a vertex \(x_t\) whose degree \(d(x_t) \neq 2\) where \(d(x_j) = 2 (j = 1,2,\ldots,t – 1)\) for every other vertex of \(P\). A hanging path \(P\) is said to be a tail of \(G\), denoted by \(t(G)\), if its length \(|t(G)|\) is a maximum among all hanging paths of \(G\). In this paper, we prove \(\zeta(T_3) = 0\), where \(T_3\) is any tree with \(|t(T_3)| \geq 3\). The result improves a previous result for integral sum trees from identification of Chen\((1998)\).
Let \(H = K_{k_1,k_2,\ldots,k_t}\) be a complete multipartite graph having \(t \geq 3\) parts. Extending the well-known result that a simple graph \(G\) or its complement, \({G}\), is connected, it is proved that in any coloring of the edges of \(H\) with two colors, blue and red, at least one of the subgraphs induced by the blue edges or by the red edges, is connected.
Given a collection of points in the plane, a circle is drawn around each point with radius equal to the smallest distance from that point to any other in the collection. The sphere-of-influence graph is the intersection graph of the open balls given by these circles. Any graph isomorphic to such a graph is a SIG realizable in a plane. Similarly, one can define a SIG realizable on a sphere by selecting a collection of points on a sphere. We show that \(K_9\) is realizable as a SIG on a sphere and that the family of graphs realizable as SIGs on a sphere is at least as large as the family of SIGs in the plane.
In this paper, we construct many Hadamard matrices of order \(44\) and we use a new efficient algorithm to investigate the lower bound of inequivalent Hadamard matrices of order \(44\). Using four \((1, -1)\)-circulant matrices of order \(11\) in the Goethals-Seidel array, we obtain many new Hadamard matrices of order \(44\) and we show that there are at least \(6018\) inequivalent Hadamard matrices for this order. Moreover, we use a known method to investigate the existence of double even self-dual codes \([88, 44, d]\) over \(\text{GF}(2)\) constructed from these Hadamard matrices.
Given positive integers \(m, k,\) and \(t\). Let \(D_{m,[k,k+i]} = \{1,2,\ldots,m\} – \{k,k+1,\ldots,k+i\}\). The distance graph \(G(\mathbb{Z}, D_{m,[k,k+i]})\) has vertex set all integers \(\mathbb{Z}\) and edges connecting \(j\) and \(j’\) whenever \(|j-j’| \in D_{m,[k,k+i]}\). The fractional chromatic number, the chromatic number, and the circular chromatic number of \(G(\mathbb{Z}, D_{m,k,i})\) are denoted by \(\chi_f(\mathbb{Z}, D_{m[k,k+i]}), \chi(\mathbb{Z}, D_{m,[k,k+i]}),\) and \(\chi_c(\mathbb{Z}, D_{m,[k,k+i]})\), respectively. For \(i=0\), we simply denote \(D_{m,[k,k+0]}\) by \(D_{m,k}\). \(X(\mathbb{Z}, D_{m,k})\) was studied by Eggleton, Erdős and Skilton [5], Kemnitz and Kolberg [8], and Liu [9], and was completely solved by Chang, Liu and Zhu [1] who also determined \(\chi_c(\mathbb{Z}, D_{m,k})\) for any \(m\) and \(k\). The value of \(\chi_c(\mathbb{Z}, D_{m,k})\) was studied by Chang, Huang and Zhu [2] who finally determined \(\chi_c(\mathbb{Z}, D_{m,k})\) for any \(m\) and \(k\). This paper extends the study of \(G(\mathbb{Z}, D_{m,[k,k+i]})\) to values \(i\) with \(1 \leq i \leq k-1\). We completely determine \(\chi_f(\mathbb{Z}, D_{m,[k,k+i]})\) and \(\chi(\mathbb{Z}, D_{m,k,i})\) for any \(m\) and \(k\) with \(1 \leq i \leq k-1\). However, for \(\chi_c(\mathbb{Z}, D_{m,[k,k+i]})\), only some special cases are determined.
We introduce graphs \(G\) with at least one maximum independent set of vertices \(I\), such that \(\forall v \in V(G) \setminus I\), the number of vertices in \(N_G(v) \cap I\) is constant. When this number of vertices is equal to \(\lambda\), we say that \(I\) has the \(\lambda\)-property and that \(G\) is \(\lambda\)-regular-stable. Furthermore, we extend the study of this property to the well-covered graphs (that is, graphs where all maximal independent sets of vertices have the same cardinality). In this study, we consider well-covered graphs for which all maximal independent sets of vertices have the \(\lambda\)-property, herein called well-covered \(\lambda\)-regular-stable graphs.
Isometric subgraphs of hypercubes are known as partial cubes. Edge-critical partial cubes are introduced as the partial cubes \(G\) for which \(G – e\) is not a partial cube for any edge \(e\) of \(G\). An expansion theorem is proved by means of which one can generate many edge-critical partial cubes. Edge-critical partial cubes are characterized among the Cartesian product graphs. We also show that the \(3\)-cube and the subdivision graph of \(K_4\) are the only edge-critical partial cubes on at most \(10\) vertices.
This paper discusses the enumeration of rooted labelled spanning forests of the complete bipartite graph \(K_{m,n}\).
A set of edges \(D\) in a graph \(G\) is a dominating set of edges if every edge not in \(D\) is adjacent to at least one edge in \(D\). The minimum cardinality of an edge dominating set of \(G\) is the edge domination number of \(G\), denoted by \(D_E(G)\). In this paper, we investigate the edge domination number for the cartesian product of an \(n\)-colorable graph \(G\) and the complete graph \(K_n\).
For graph \(G\) with non-empty edge set, a \((j,k)\)-edge labeling of \(G\) is an integer labeling of the edges such that adjacent edges receive labels that differ by at least \(j\), and edges which are distance two apart receive labels that differ by at least \(k\). The \(\lambda’_{j,k}\)-number of \(G\) is the minimum span over the \((j,k)\)-edge labelings of \(G\). By establishing the equivalence of the edge labelings of \(G\) to particular vertex labelings of \(G\) and the line graph of \(G\), we explore the properties of \(\chi_{j,k}(G)\). In particular, we obtain bounds on \(\lambda’_{j,k}(G)\), and prove that the \(\Delta^2\) conjecture of Griggs and Yeh is true for graph \(H\) if \(H\) is the line graph of some graph \(G\). We investigate the \(\lambda’_{1,1}\)-numbers and \(\lambda_{2,1}\)-numbers of common classes of graphs, including complete graphs, trees, \(n\)-cubes, and joins.
The \(n \times n\) Lah matrix \(L_n\) is defined by \((L_n)_{ij} = l(i, j)\), where \({l}(i, j)\) is the unsigned Lah number. In this paper, we investigate the algebraic properties of \(L_n\), and many important relations between \({L}_n\) and Pascal matrix and Stirling matrix, respectively. In addition, we obtain its exponential expansion and Pascal matrix factorization. Furthermore, we introduce a simple method to find and prove combinatorial identities.
Let \(K_n\) be the complete graph on \(n\) vertices. In this paper, we find the necessary and sufficient conditions for the existence of an \((m_1, m_2, \ldots, m_r)\)-cycle system of \(K_n\), where \(m_i\) (\(1 \leq i \leq r\)) are positive even integers, and \(\sum_{i=1}^{r}m_i = 2^k\) for \(k \geq 2\). In particular, if \(r = 1\) then there exists a cyclic \(2^k\)-cycle system of \(K_n\) if and only if \(2^k\) divides \(|E(K_n)|\) and \(n\) is odd.
In 1948, de Bruijn and Erdős proved that every finite linear space on \(v\) points and with \(6\) lines fulfils the inequality \(b \geq v\), and the equality holds if the linear space is a (possibly degenerate) projective plane. This result led to the problem of classifying finite linear spaces on \(v\) points and with \(b = v + s\) lines, \(s \geq 1\). This paper contains the classification of finite linear spaces on \(v\) points and with \(b = v + 4\) lines.
For a vertex \(v\) of a connected graph \(G\) and a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v,S) = \min\{d(v,z)|z \in S\}\). For an ordered \(k\)-partition \(\Pi = \{S_1,S_2,\ldots,S_k\}\) of \(V(G)\), the code of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(c_\Pi(v) = (d(v, S_1), d(v, S_2), \ldots, d(v,S_k))\). The \(k\)-partition \(\Pi\) is a resolving partition if the \(k\)-vectors \(c_\Pi(v), v \in V(G)\), are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is the partition dimension \(pd(G)\) of \(G\). A resolving partition \(\Pi = \{S_1,S_2,\ldots,S_k\}\) of \(V(G)\) is a resolving-coloring if each \(S_i\) (\(1 \leq i \leq k\)) is independent and the resolving-chromatic number \(\chi_r(G)\) is the minimum number of colors in a resolving-coloring of \(G\). A resolving partition \(\Pi = \{S_1,S_2,\ldots,S_k\}\) is acyclic if each subgraph \((S_i)\) induced by \(S_i\) (\(1 \leq i \leq k\)) is acyclic in \(G\). The minimum \(k\) for which there is a resolving acyclic \(k\)-partition of \(V(G)\) is the resolving acyclic number \(\alpha_r(G)\) of \(G\). Thus \(2 \leq pd(G) < \alpha_r(G) \leq \chi_r(G) \leq n\) for every connected graph \(G\) of order \(n \geq 2\). We present bounds for the resolving acyclic number of a connected graph in terms of its arboricity, partition dimension, resolving-chromatic number, diameter, girth, and other parameters. Connected graphs of order \(n \geq 3\) having resolving acyclic number \(2, n,\) or \(n-1\) are characterized.
Let \(p\) and \(q\) be distinct primes with \(p > q\) and \(n\) a positive integer. In this paper, we consider the set of possible cross numbers for the cyclic groups \(\mathbb{Z}_{2p^n}\) and \(\mathbb{Z}_{pq}\). We completely determine this set for \(\mathbb{Z}_{2p^n}\) and also \(\mathbb{Z}_{pq}\) for \(q = 3, q = 5\) and the case where \(p\) is sufficiently larger than \(g\). We view the latter result in terms of an upper bound for this set developed in a paper of Geroldinger and Schneider [8] and show precisely when this upper bound is an equality.
It is known that triangles with vertices in the integral lattice \(\mathbb{Z}^2\) and exactly one interior lattice point can have \(3, 4, 6, 8\), and \(9\) lattice points on their boundaries. No such triangles with \(5\), nor \(7\), nor \(n \geq 10\) boundary lattice points exist. The purpose of this note is to study an analogous property for Hex-triangles, that is, triangles with vertices in the set \(H\) of corners of a tiling of \(\mathbb{R}^2\) by regular hexagons of unit edge. We show that any Hex-triangle with exactly one interior \(H\)-point can have \(3, 4, 5, 6, 7, 8,\) or \(10\), \(H\)-points on its boundary and cannot have \(9\) nor \(n \geq 11\) such points.
The problem of classification of Hadamard matrices becomes an NP-hard problem as the order of the Hadamard matrices increases. In this paper, we use a new criterion which inspired us to develop an efficient algorithm to investigate the lower bound of inequivalent Hadamard matrices of order \(36\). Using four \((1,-1)\) circulant matrices of order \(9\) in the Goethals-Seidel array, we obtain many new Hadamard matrices of order \(36\) and we show that there are at least \(1036\) inequivalent Hadamard matrices for this order.
We prove the gracefulness of two classes of graphs.
Let \(G\) be a graph with \(q\) edges. \(G\) is numbered if each vertex \(v\) is assigned a non-negative integer \(\phi(v)\) and each edge \(uv\) is assigned the value \(|\phi(u) – \phi(v)|\). The numbering is called graceful if, further, the vertices are labelled with distinct integers from \(\{0, 1, 2, \ldots, q\}\) and the edges with integers from \(1\) to \(q\). A graph which admits a graceful numbering is said to be graceful. For the literature on graceful graphs see [1, 2] and the relevant references given in them.
Let \(G\) be a graph and let \(c\) be a coloring of its edges. If the sequence of colors along a walk of \(G\) is of the form \(a_1, \ldots, a_n, a_1, \ldots, a_n\), the walk is called a square walk. We say that the coloring \(c\) is square-free if any open walk is not a square and call the minimum number of colors needed so that \(G\) has a square-free coloring a walk Thue number and denote it by \(\pi_w(G)\). This concept is a variation of the Thue number introduced by Alon, Grytczuk, Hatuzczak, and Riordan in [2].
Using the walk Thue number, several results of [1] are extended. The Thue number of some complete graphs is extended to Hamming graphs. This result (for the case of hypercubes) is used to show that if a graph \(G\) on \(n\) vertices and \(m\) edges is the subdivision graph of some graph, then \(\pi_w(G) \leq n – \frac{m}{2}\). Graph products are also considered. An inequality for the Thue number of the Cartesian product of trees is extended to arbitrary graphs and upper bounds for the (walk) Thue number of the direct and the strong products are also given. Using the latter results, the (walk) Thue number of complete multipartite graphs is bounded, which in turn gives a bound for arbitrary graphs in general and for perfect graphs in particular.
In the paper [3], the theorem that at least \( \frac{n – 1}{2} \) queens are required to dominate the \( n \times n \) chessboard was attributed to P. H. Spencer, in [1]. A proof of this result appeared in the earlier work [2].
A set \( D \) of vertices in a graph \( G \) is irredundant if every vertex \( v \) in \( D \) has at least one private neighbour in \( N[v, G] \setminus N[D \setminus \{v\}, G] \). A set \( D \) of vertices in a graph \( G \) is a minimal dominating set of \( G \) if \( D \) is irredundant and every vertex in \( V(G) \setminus D \) has at least one neighbour in \( D \). Further, irredundant sets and minimal dominating sets of maximal cardinality are called \( IR \)-sets and \( \Gamma \)-sets, respectively. A set \( I \) of the vertex set of a graph \( G \) is independent if no two vertices in \( I \) are adjacent, and independent sets of maximal cardinality are called \( \alpha \)-sets.
In this paper, we prove that bipartite graphs and chordal graphs have a unique \( \alpha \)-set if and only if they have a unique \( \Gamma \)-set if and only if they have a unique \( IR \)-set. Some related results are also presented.
Static mastermind is like normal mastermind, except that the codebreaker must supply at one go a list of questions (candidate codes), the answers to which must uniquely determine the secret code. We confirm the minimum size list for some small values. Then we solve the game for up to 4 positions. In particular, we show that for \( k \) sufficiently large, the minimum size of a list for 4 positions and \( k \) colours is \( k – 1 \).
It is shown that for \( n \geq 16 \), the sum of cardinalities of open irredundant sets in an \( n \)-vertex graph and its complement is at most \( \frac{3n}{4} \).
The redundance \( R(G) \) of a graph \( G \) is the minimum, over all dominating sets \( S \), of \( \sum_{v \in S} (1 + \deg(v)) \), where \( \deg(v) \) is the degree of vertex \( v \). We use some dynamic programming algorithms to compute the redundance of complete grid graphs \( G_{m,n} \) for \( 1 \leq m \leq 21 \) and all \( n \), and to establish good upper and lower bounds on the redundance for larger \( m \). We conjecture that the upper bound is the redundance when \( m > 21 \).
Heinrich et al. [4] characterized those simple eulerian graphs with no Petersen-minor which admit a triangle-free cycle decomposition, a TFCD. If one permits Petersen minors then no such characterization is known even for \( {E}(4,2) \), the set of all the eulerian graphs of maximum degree 4. Let \( {EM}(4,2) \subset {E}(4,2) \) be the set of all graphs \( H \) such that all triangles of \( H \) are vertex disjoint, and each triangle contains a degree 2 vertex in \( H \). In the paper it is shown that to each \( G \in {E}(4,2) \) there exists a finite subset \( S \subset {EM}(4,2) \) so that \( G \) admits a TFCD if and only if some \( H \in S \) admits a TFCD. Further, some sufficient conditions for a graph \( G \in {E}(4,2) \) to possess a TFCD are given.
Let \( \nu \) be some graph parameter and let \( \mathcal{G} \) be a class of graphs for which \( \nu \) can be computed in polynomial time. In this situation, it is often possible to devise a strategy to decide in polynomial time whether \( \nu \) has a unique realization for some graph in \( \mathcal{G} \). We first give an informal description of the conditions that allow one to devise such a strategy, and then we demonstrate our approach for three well-known graph parameters: the domination number, the independence number, and the chromatic number.
A \( k \)-line-distinguishing coloring of a graph \( G = (V, E) \) is a partition of \( V \) into \( k \) sets \( V_1, \ldots, V_k \) such that \( q(\langle V_i \rangle) \leq 1 \) for \( i = 1, \ldots, k \) and \( q(V_i, V_j) \leq 1 \) for \( 1 \leq i \leq j \leq k \). If the color classes in a line-distinguishing coloring are also independent, then it is called a harmonious coloring. A coloring is minimal if, when two color classes are combined, we no longer have a coloring of the given type.
The upper harmonious chromatic number, \( H(G) \), is defined as the maximum cardinality of a minimal harmonious coloring of a graph \( G \), while the upper line-distinguishing chromatic number, \( H'(G) \), is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \( G \). For any graph \( G \) of maximum degree \( \Delta(G) \), \( H'(G) \geq \Delta(G) \) and \( H(G) \geq \Delta(G) + 1 \).
We characterize connected graphs \( G \) that contain neither a triangle nor a 5-cycle for which \( H(G) = \Delta(G) + 1 \). We show that a triangle-free connected graph \( G \) satisfies \( H'(G) = \Delta(G) \) if and only if \( G \) is a star \( K_{1, \Delta(G)} \). A partial characterization of connected graphs \( G \) for which \( H'(G) = \Delta(G) \) is obtained.
There are at least 52432 symmetric \( (100, 45, 20) \) designs on which \( \text{Frob}_{10} \times \mathbb{Z}_2 \) acts as an automorphism group. All these designs correspond to Bush-type Hadamard matrices of order 100, and each leads to an infinite class of twin designs with parameters
\[v= 100(81^m + 81^{m-1} + \ldots + 81+1),\, k=45(81)^m ,\, \lambda=20(81)^m ,\]
and an infinite class of Siamese twin designs with parameters
\[v= 100(121^m + 121^{m-1} + \ldots + 121+1),\, k=55(121)^m ,\, \lambda=30(121)^m ,\]
where \( m \) is an arbitrary positive integer. One of the constructed designs is isomorphic to that used by Z. Janko, H. Kharaghani, and V. D. Tonchev [4].
We define the \( B_2 \) block-intersection graph of a balanced incomplete block design \( (V,\mathfrak{B}) \) having order \( n \), block size \( k \), and index \( \lambda \), or BIBD\( (n,k,\lambda) \), to be the graph with vertex set \( \mathfrak{B} \) in which two vertices are adjacent if and only if their corresponding blocks have exactly two points of \( V \) in common. We define an undirected (resp. directed) hinge to be the multigraph with four vertices which consists of two undirected (resp. directed) 3-cycles which share exactly two vertices in common. An undirected (resp. directed) hinge system of order \( n \) and index \( \lambda \) is a decomposition of \( \lambda K_n \) (resp. \( \lambda{K}_n^* \)) into undirected (resp. directed) hinges. In this paper, we show that each component of the \( B_2 \) block-intersection graph of a simple BIBD\( (n,3,2) \) is 2-edge-connected; this enables us to decompose pure Mendelsohn triple systems and simple 2-fold triple systems into directed and undirected hinge systems, respectively. Furthermore, we obtain a generalisation of the Doyen-Wilson theorem by giving necessary and sufficient conditions for embedding undirected (resp. directed) hinge systems of order \( n \) in undirected (resp. directed) hinge systems of order \( v \). Finally, we determine the spectrum for undirected hinge systems for all indices \( \lambda \geq 2 \) and for directed hinge systems for all indices \( \lambda \geq 1 \).
Vince asked whether for each rational \( r \) between 2 and 4 there was a planar graph of circular chromatic number \( r \). Moser and Zhu showed that the answer is yes, the first for \( 2 < r < 3 \), the second for \( 3 < r < 4 \). This paper gives another family of planar graphs with circular chromatic number between 2 and 3.
We present a new proof that the optimal fast solutions to the gossip problem, for an even number of participants \( n > 2^{\lceil \log_2{n} \rceil} – 2^{\lfloor \lceil \log_2{n} \rceil /2\rfloor} \), require exactly \( \frac{n}{2}\lceil \log_2{n} \rceil \) calls.
We establish that up to an isomorphism there are exactly \(88\) perfect \(1\)-factorizations of \( K_{16} \) having nontrivial automorphism group. We also present some related results.
We consider the firefighter problem. We begin by proving that the associated decision problem is NP-complete even when restricted to bipartite graphs. We then investigate algorithms and bounds for trees and square grids.
Face two-colourable triangular embeddings of complete graphs \(K_n\) correspond to biembeddings of Steiner triple systems. Such embeddings exist only if \( n \) is congruent to 1 or 3 modulo 6. In this paper, we present the number of these embeddings for \( n = 13 \).
The trace of a degree \( n \) polynomial \( p(x) \) over \( \text{GF}(2) \) is the coefficient of \( x^{n-1} \), and the \({subtrace}\) is the coefficient of \( x^{n-2} \). We derive an explicit formula for the number of irreducible degree \( n \) polynomials over \( \text{GF}(2) \) that have a given trace and subtrace. The trace and subtrace of an element \( \beta \in \text{GF}(2^n) \) are defined to be the coefficients of \( x^{n-1} \) and \( x^{n-2} \), respectively, in the polynomial \(q(x) = \prod_{i=0}^{n-1} (x + \beta^{2^i}).\) We also derive an explicit formula for the number of elements of \( \text{GF}(2^n) \) of given trace and subtrace. Moreover, a new two-equation Möbius-type inversion formula is proved.
In this paper, it has been verified, by a computer-based proof, that the smallest size of a complete arc is 12 in \( \text{PG}(2,27) \) and 13 in \( \text{PG}(2,29) \). Also, the spectrum of the sizes of the complete arcs of \( \text{PG}(2,27) \) has been found. The classification of the smallest complete arcs of \( \text{PG}(2,27) \) is given: there are seven non-equivalent 12-arcs, and for each of them, the automorphism group and some geometrical properties are presented. Some examples of complete 13-arcs of \( \text{PG}(2,29) \) are also described.
For a factorization \( F \) of a graph \( G \) into factors \( F_1, F_2, \ldots, F_k \), the chromatic number \( \chi(F) \) of \( F \) is the minimum number of elements \( V_1, V_2, \ldots, V_m \) in a partition of \( V(G) \) such that each subset \( V_i \) \((1 \leq i \leq m)\) is independent in some factor \( F_j \) \((1 \leq j \leq k)\). If \( \chi(F) = m \), then \( F \) is an \( m \)-chromatic factorization. For integers \( k, m, n \geq 2 \) with \( n \geq m \), the cofactor number \( c_m(k,n) \) is defined as the smallest positive integer \( p \) for which there exists an \( m \)-chromatic factorization \( F \) of the complete graph \( K_p \) into \( k \) factors \( F_1, F_2, \ldots, F_k \) such that \( \chi(F_i) \geq n \) for all integers \( i \) \((1 \leq i \leq k)\). The values of the numbers \( c_m(k,n) \) are investigated for \( m = 3 \) and \( m = 4 \).The \( k \)-cofactorization number \( \chi_k(G) \) of a graph \( G \) is defined as \( \max\{\chi(F) : F \text{ is a factorization of } G \text{ into } k \text{ factors}\} \). It is shown that \( \chi_k(K_n) \geq \lfloor n^{1/k} \rfloor \) for \( k \geq 2 \) and \( n \geq 4 \). The numbers \( \chi_k(K_n) \) are determined for several values of \( k \) and \( n \).
Denote the total domination number of a graph \(G\) by \(\gamma_t(G)\). A graph \(G\) is said to be total domination edge critical, or simply \(\gamma_t\)-critical, if \(\gamma_t(G+e) < \gamma_t(G)\) for each edge \(e \in E(\overline{G})\). For \(\gamma_t\)-critical graphs \(G\), that is, \(\gamma_t\)-critical graphs with \(\gamma_t(G) = 3\), the diameter of \(G\) is either \(2\) or \(3\). We study the \(3_t\)-critical graphs \(G\) with \(diam(G) = 2\).
We consider two possible methods of embedding a (simple undirected) graph into a uniquely vertex colourable graph. The first method considered is to build a \(K\)-chromatic uniquely vertex colourable graph from a \(k\)-chromatic graph \(G\) on \(G\cup K_k\), by adding a set of new edges between the two components. This gives rise to a new parameter called fixing number (Daneshgar (1997)). Our main result in this direction is to prove that a graph is uniquely vertex colourable if and only if its fixing number is equal to zero (which is a counterpart to the same kind of result for defining numbers proved by Hajiabolhassan et al. (1996)).
In our second approach, we try a more subtle method of embedding which gives rise to the parameters \(t_r\)-fixer and \(\tau_r\)-index (\(r = 0, 1\)) for graphs. In this approach we show the existence of certain classes of \(u\)-cores, for which, the existence of an extremal graph provides a counter example for Xu’s conjecture.
Necessary and sufficient conditions are given for a Steiner triple system of order \(t\) admitting an automorphism consisting of one large cycle, cycles of length \(8\), and a fixed point, with \(t \leq 4\). Necessary conditions are given for all \(t \geq 1\).
In this note we prove that the bipartite Ramsey number for \(K_{2,n}\) with \(q\) colors does not exceed \((n-1)q^2+q+1-\left\lceil\sqrt{q}\right\rceil\), improving the previous upper bound by \(\left\lceil\sqrt{q}\right\rceil-2\).
Let \(\delta_k\) denote the minimum degree of the \(k^{th}\)-iterated line graph \(L^k(G)\). For any connected graph \(G\) that is not a path, the inequality \(\delta_k \geq 2\delta_k – 2\) holds. Niepel, Knor, and Soltés [5] have conjectured that there exists an integer \(K\) such that, for all \(k \geq K\), equality holds; that is, the minimum degree \(\delta_k\) attains the least possible growth. We prove this conjecture by extending the methods we used in [2] for a similar conjecture about the maximum degree.
A set of Knights covers a board if a Knight attacks every unoccupied square. What is the minimum number of Knights in a cover of an \(n\times n\) board? For \(n \leq 10\), we give a non-computational proof that the widely accepted answers are correct. For \(n \leq 14\), fractional Knight packings are used in an exhaustive branch-and-bound program. This gives the first enumeration of minimum Knight covers for \(11 \leq n \leq 14\). For \(n \geq 15\), integer programs are used to find small (though not necessarily minimum) symmetric covers. This yields smaller covers for \(16 \leq n \leq 19\), and new covers when \(21 \leq n \leq 25\). Simulated annealing discovered yet smaller covers for \(n = 19\) and \(n = 21\). Guess work improved the results for \(n = 20\) and \(n = 25\).
It has been shown that if \(G = (V, E)\) is a simple graph with \(n\) vertices, \(m\) edges, an average (per edge) of \(t\) triangles occurring on the edges, and \(J = \max_{uv \in E} |N(u) \cup N(v)|\), then \(4m \leq n(J+t)\). The extremal graphs for this inequality for \(J = n\) and \(J = n – 1\) have been determined. For \(J = n\), the extremal graphs are the Turán graphs with parts of equal size; notice that these are the complements of the strongly regular graphs with \(\mu = 0\). For \(J = n-1\), the extremal graphs are the complements of the strongly regular graphs with \(\mu = 1\). (The only such graphs known to exist are the Moore graphs of diameter \(2\)).
For \(J = n-2\) and \(t = 0\), it has recently been shown that the only extremal graph (except when \(n = 8, 10\)) is \(K_{n/2,n/2} – (1\text{-factor})\). Here, we use a well-known theorem of Andrásfai, Erdős, and Sós to characterize the extremal graphs for \(t = 0\), any given value of \(n-J\), and \(n\) sufficiently large (they are the regular bipartite graphs). Then we give some examples of extremal non-bipartite graphs for smaller values of \(n\).
Let \(G = (V, E)\) be a graph. Let \(\Phi: V \to {R}\), where \({R}\) is the set of all reals (\({R}\) can be replaced by any chain). We say that \(u\) \(\Phi\)-strongly dominates \(v\) and \(v\) \(\Phi\)-weakly dominates \(u\) if \(uv \in E\) and \(\Phi(u) \geq \Phi(v)\). When \(\Phi\) is a constant function, we have the usual domination and when \(\Phi\) is the degree function of the graph, we have the strong (weak) domination studied by Sampathkumar et al. In this paper, we extend the results of O. Ore regarding minimal dominating sets of a graph. We also extend the concept of fully domination balance introduced by Sampathkumar et al and obtain a lower bound for strong domination number of a graph.
A function \(f: V \to \{-1, 1\}\) defined on the vertices of a graph \(G = (V, E)\) is a signed \(2\)-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every \(v \in V\), \(f(N[v]) \leq 1\), where \(N[v]\) consists of \(v\) and every vertex adjacent to \(v\). The weight of a signed \(2\)-independence function is \(f(V) = \sum f(v)\). The signed \(2\)-independence number of a graph \(G\), denoted \(\alpha^2_s(G)\), is the maximum weight of a signed \(2\)-independence function of \(G\). In this article, we give some new upper bounds on \(\alpha^2_s(G)\) of \(G\), and establish a sharp upper bound on \(\alpha^2_s(G)\) for an \(r\)-partite graph.
Let \(G\) be a graph with a perfect matching \(M_0\). It is proved that \(G\) is \(1\)-extendable if and only if for any pair of vertices \(x\) and \(y\) with an \(M_0\)-alternating path \(P_y\) of length three which starts with an edge that belongs to \(M_0\), there exists an \(M_0\)-alternating path \(P\) connecting \(x\) and \(y\), of which the starting and the ending edges do not belong to \(M_0\). With this theorem, we develop a polynomial algorithm that determines whether the input graph \(G\) is \(1\)-extendable, the time complexity of the algorithm is \(O(|E|^2)\).
Let \(G\) be a graph on \(p\) vertices and denote by \(L(G) = D(G) – A(G)\) the difference between the diagonal matrix of vertex degrees and the adjacency matrix. It is not difficult to see that \(L(G)\) is positive semidefinite symmetric and its second smallest eigenvalue, \(a(G) > 0\), if and only if \(G\) is connected. This observation led M. Fiedler to call \(a(G)\) the algebraic connectivity of \(G\).
The algebraic connectivity of the line graph, the middle graph, and the total graph of a regular graph are given.
The minimum number of incomplete blocks required to cover, exactly \(A\) times, all \(t\)-element subsets from a set \(V\) of cardinality \(v\) (\(v \geq t\)) is denoted by \(g(\lambda,t;v)\). The value of \(g(2,2;v)\) is known for \(v = 3, 4, \ldots, 11\). It was previously known that \(14 \leq g(2, 2; 12) \leq 16\). We prove that \(g(2,2;12) \geq 15\).
A computer program for finding knight coverings of a chessboard is described, and some improved coverings for boards of sizes \(16\times 16\) through \(25\times 25\) are shown.
Let \(G\) be a graph and \(d, d’\) be positive integers, \(d’ \geq d\). An \(m\)-\((d, d’)\)-circular distance two labeling is a function \(f\) from \(V(G)\) to \(\{0, 1, 2, \ldots, m-1\}\) such that:\(|f(u) – f(v)|_m \geq d\) if \(u\) and \(v\) are adjacent; and \(|f(u) – f(v)|_m \geq d’\) if \(u\) and \(v\) are distance two apart, where \(|x|_m := \min\{|x|, m – |x|\}\) .The minimum \(m\) such that there exists an \(m\)-\((d, d’)\)-circular labeling for \(G\) is called the \(\sigma_{d, d’}\)-number of \(G\) and denoted by \(\sigma_{d, d’}(G)\). The \(\sigma_{d, d’}\)-numbers for trees can be obtained by a first-fit algorithm. In this article, we completely determine the \(\sigma_{d, 1}\)-numbers for cycles. In addition, we show connections between generalized circular distance labeling and circular chromatic number.
The inventory of a \(2 \times m\) array \(A = A(i,j)\) consisting of \(n\) not necessarily distinct positive integers \(\mathbb{I}(2,j)\) is the \(2 \times n\) array \(\mathbb{I}(A) = \mathbb{I}(i,j)\), where \(\mathbb{I}(i,j)\) is the number of occurrences of \(\mathbb{I}(1,j)\) in \(A\). Define \(\mathbb{I}^q(A) = I(\mathbb{I}^{q-1}(A))\) for \(q \geq 1\), with \(\mathbb{I}^0(A) = A\). For every \(A\), the chain \(\{\mathbb{I}^q(A)\}\) of inventories is eventually periodic, with period \(1, 2\), or \(3\). The proof depends on runlengths of partitions of integers. A final section is devoted to an open question about cumulative inventory chains.
A decomposition \(\mathcal{F} = \{F_1, \ldots, F_r\}\) of the edge set of a graph \(G\) is called a resolving \(r\)-decomposition if for any pair of edges \(e_1\) and \(e_2\), there exists an index \(i\) such that \(d(e_1, F_i) \neq d(e_2, F_i)\), where \(d(e, F)\) denotes the distance from \(e\) to \(F\). The decomposition dimension \(dec(G)\) of a graph \(G\) is the least integer \(r\) such that there exists a resolving \(r\)-decomposition. Let \(K_n\) be the complete graph with \(n\) vertices. It is proved that \(dec(K_n) \leq \frac{1}{2} (\log_2 n)^2 (1 + o(1)).\)
For a vertex \(v\) of a graph \(G = (V, E)\), the lower independence number \(i_v(G)\) of \(G\) relative to \(v\) is the minimum cardinality of a maximal independent set in \(G\) that contains \(v\). The average lower independence number of \(G\) is \(i_{av}(G) = \frac{1}{|V|} \sum_{v\in V} i_v(G)\). In this paper, we show that if \(G\) is a tree of order \(n\), then \(i_{av}(G) \geq {2}\sqrt{n} + O(1)\), while if \(G\) is an outer-planar graph of order \(n\), then \(i_{av}(G) \geq 2\sqrt{\frac{n}{3}} + O(1)\). Both bounds are asymptotically sharp.
We consider the partition function \(b’_p(n)\), which counts the number of partitions of the integer \(n\) into distinct parts with no part divisible by the prime \(p\). We prove the following: Let \(p\) be a prime greater than \(3\) and let \(r\) be an integer between \(1\) and \(p-1\), inclusively, such that \(24r+1\) is a quadratic nonresidue modulo \(p\). Then, for all nonnegative integers \(n\), \(b’_p{(pn+r)} \equiv 0 \pmod{2}.\)
We show that:(a) the special product of two cycles is Hamiltonian decomposable, and (b) if \(G_1\) and \(G_2\) are two Hamiltonian decomposable graphs and at least one of their complements is Hamiltonian decomposable, then the special product of \(G_1\) and \(G_2\) is Hamiltonian decomposable.
A vertex-magic total labeling on a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(1, 2, \ldots, |V(G) \cup E(G)|\) with the property that, given any vertex \(x\), \(\lambda(x) + \sum_{y \sim x} \lambda(y) = k\) for some constant \(k\).
In this paper, we completely determine which complete bipartite graphs have vertex-magic total labelings.
In this paper, the notions of \(c\)-Motzkin and \(d\)-Motzkin words are introduced, studied, and the cardinal numbers of their sets are evaluated. Finally, bijections between the sets of the introduced Motzkin words and certain sets of noncrossing partitions are exhibited.
Vizing conjectured that \(\gamma(G)\gamma(H) \leq \gamma(G \Box H)\) for all graphs \(G\) and \(H\), where \(\gamma(G)\) denotes the domination number of \(G\) and \(G \Box H\) is the Cartesian product of \(G\) and \(H\). We prove that if \(G\) and \(H\) are \(\delta\)-regular, then, with only a few possible exceptions, Vizing’s conjecture holds. We also prove that if \(\delta(G), \Delta(G), \delta(H)\), and \(\Delta(H)\) are in a certain range, then Vizing’s conjecture holds. In particular, we show that for graphs of order at most \(n\) with minimum degrees at least \(\sqrt{n} \ln n\), the conjecture holds.
The classification of Hadamard matrices of orders \(n \geq 32\) remains an open and difficult problem. The definition of equivalent Hadamard matrices gets increasingly complex as \(n\) grows larger. One efficient criterion (\(K\)-boxes) has been used for the construction of inequivalent Hadamard matrices in order \(28\).
In this paper, we use inequivalent projections of Hadamard matrices and their symmetric Hamming distances to check for inequivalent Hadamard matrices. Using this criterion, we have developed two algorithms. The first one achieves finding all inequivalent projections in \(k\) columns as well as classifying Hadamard matrices, and the second, which is faster than the first, uses the symmetric Hamming distance distribution of projections to classify Hadamard matrices. As an example, we apply the second algorithm to the known inequivalent Hadamard matrices of orders \(n = 4, 8, 12, 16, 20, 24\), and \(28\).
A composition of a positive integer \(n\) consists of an ordered sequence of positive integers whose sum is \(n\). A palindromic composition is one for which the sequence is the same from left to right as from right to left. This paper shows various ways of generating all palindromic compositions, counts the number of times each integer appears as a summand among all the palindromic compositions of \(n\), and describes several patterns among the numbers generated in the process of enumeration.
The study of the maximum size \(ex(n; K_{t,t})\) of a graph of order \(n\) not containing the complete bipartite graph \(K_{t,t}\) as a subgraph is the aim of this paper. We show an upper bound for this extremal function that is optimum for infinitely many values of \(n\) and \(t\). Moreover, we characterize the corresponding family of extremal graphs.
In this paper we extend the work of Bogart and Trenk [3] and Fishburn and Trotter [6] in studying different classes of bitolerance orders. We provide a more comprehensive list of classes of bitolerance orders and prove equality between some of these classes in general and other classes in the bipartite domain. We also provide separating examples between unequal classes of bitolerance orders.
We consider non-crossing trees and show that the height of node \(\rho n\) with \(0 < p < 1\) in a non-crossing tree of size \(n\) is asymptotically Maxwell-distributed. We also give an asymptotic formula for the expected height of node \(\rho n\).
Let \(G = (V(G), E(G))\) be a finite simple graph with \(p\) vertices and \(n\) edges. A labeling of \(G\) is an injection \(f: V(G) \to {Z}_n\). A labeling of \(G\) is called \(2\)-sequential if \(f(V(G)) = \{r, r+1, \ldots, r+p-1\}\) (\(0 \leq r <r+ p-1 \leq n-1\)) and the induced edge labeling \(f^*: E(G) \to \{0, 1, \ldots, n-1\}\) given by \[f^*(u,v) = f(u) + f(v), \quad \text{for every edge } (u,v) \] forms a sequence of distinct consecutive integers \(\{k, k+1, \ldots, n+k-1\}\) for some \(k\) (\(1 \leq k \leq n-2\)). By utilizing the graphs having \(2\)-sequential labeling, several new families of sequential graphs are presented.
A cycle \(C\) in a graph \(G\) is said to be a dominating cycle if every vertex of \(G\) has a neighbor on \(C\). Strengthening a result of Bondy and Fan [3] for tough graphs, we prove that a \(k\)-connected graph \(G\) (\(k \geq 2\)) of order \(p\) with \(t(G) > \frac{k}{k+1}\) has a dominating cycle if \(\sum_{x \in S} \geq p – 2k – 2\) for each \(S \subset V(G)\) of order \(k+1\) in which every pair of vertices in \(S\) have distance at least four in \(G\).
Let \( T \) be a partial Latin square. If there exist two distinct Latin squares \( M \) and \( N \) of the same order such that \( M \cap N = T \), then \( M \setminus T \) is said to be a Latin trade. For a given Latin square \( M \), it is possible to identify a subset of entries, termed a critical set, which intersects all Latin trades in \( M \) and is minimal with respect to this property.
Stinson and van Rees have shown that under certain circumstances, critical sets in Latin squares \( M \) and \( N \) can be used to identify critical sets in the direct product \( M \times N \). This paper presents a refinement of Stinson and van Rees’ results and applies this theory to prove the existence of two new families of critical sets.
We obtain necessary conditions for the enclosing of a group divisible design with block size 3, \( \text{GDD}(n, m; \lambda) \), into a group divisible design \( \text{GDD}(\text{n}, \text{m+1}; \lambda+\text{x}) \) with one extra group and minimal increase in \( \lambda \). We prove that the necessary conditions are sufficient for the existence of all such enclosings for GDDs with group size 2 and \( \lambda \leq 6 \), and for any \( \lambda \) when \( v \) is sufficiently large relative to \( \lambda \).
A known convolution identity involving the Catalan numbers is presented and discussed. Catalan’s original formulation, which is algebraically straightforward, is similar in style to one reported previously by the first author and the result has some interesting combinatorial aspects.
In this paper we prove various properties of the meanders. We then use these properties in order to construct recursively the set of all meanders of any particular order.
The main results of this paper are the discovery of infinite families of flow equivalent pairs of \( B_5 \) and \( W_5 \), amalamorphs, and infinite families of chromatically equivalent pairs of \( P \) and \( W_5^* \); homeomorphs, where \( B_5 \) is \( K_5 \) with one edge deleted, \( P \) is the Prism graph, and \( W_5 \) is the join of \( K_1 \) and a cycle on 4 vertices. Six families of \( B_5 \) amalamorphs, with two families having 6 parameters, and 9 families of \( W_5 \) amalamorphs, with one family having 4 parameters, are discovered. Since \( B_5 \) and \( W_5 \) are both planar, all these results obtained can be stated in terms of chromatically equivalent pairs of \( B_5^* \) and \( W_5^* \) homeomorphs. Also, three conjectures are made about the non-existence of flow-equivalent amalamorphs or chromatically equivalent homeomorphs of certain graphs.
Agrawal provided a construction for designs for two-way elimination of heterogeneity, based on a symmetric balanced incomplete block design. He could not prove the construction, although he found no counterexample. Subsequently Raghavarao and Nageswarerao published a proof of the method. In this note we observe a flaw in the published proof.
We discuss van der Waerden’s theorem on arithmetic progressions and an extension using Ramsey’s theorem, and the canonical versions. We then turn to a result (Theorem 6 below) similar in character to van der Waerden’s theorem, applications of Theorem 6, and possible canonical versions of Theorem 6. We mention several open questions involving arithmetic progressions and other types of progressions.
Let \( c^* = \). If we remove the double edge, the result is a \( 4 \)-cycle. Let \( (S,T) \) be a \( 2 \)-fold triple system without repeated triples and \( (S,C^*) \) a pairing of the triples into copies of \( c^* \). If \( C \) is the collection of \( 4 \)-cycles obtained by removing the double edges from each copy of \( c^* \) and \( F \) is a reassembly of these double edges into \( 4 \)-cycles, then \( (S,C \cup F) \) is a \( 2 \)-fold \( 4 \)-cycle system. We show that the spectrum for \( 2 \)-fold triple systems having a \({metamorphosis}\) into a \( 2 \)-fold \( 4 \)-cycle system as described above is precisely the set of all \( n \equiv 0,1,4\, \text{or}\, 9 \pmod{12} \geq 5 \).
Consider a graph \( G \) in which the vertices are partitioned into \( k \) subsets. For each subset, we want a set of vertices of \( G \) that dominate that subset. Note that the vertices doing the domination need not be in the subset itself. We are interested in dominating the entire graph \( G \) as well as dominating each of the \( k \) subsets and minimizing the sum of these \( k + 1 \) dominating sets. For trees and for all values of \( k \), we can determine an upper bound on this sum and characterize the trees that achieve it.
A technique is described that constructs a 4-colouring of a planar triangulation in quadratic time. The method is based on iterating Kempe’s technique. The heuristic gives rise to an interesting family of graphs which cause the algorithm to cycle. The structure of these graphs is described. A modified algorithm that appears always to work is presented. These techniques may lead to a proof of the 4-Colour Theorem which does not require a computer to construct and colour irreducible configurations.
For \( k > 0 \), we call a graph \( G = (V,E) \) \( k \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_k^* \) such that the induced vertex set labeling \( l^+: V(G) \to \mathbb{Z}_k \), defined by \(l^+(v) = \sum\{l(u,v): (u,v) \in E(G)\}\) is a constant map. We denote the set of all \( k \) such that \( G \) is \( k \)-magic by \(\text{IM}(G)\). We call this set the \textbf{\emph{integer-magic spectrum}} of \( G \). We investigate these sets for trees, double trees, and abbreviated double trees. We define group-magic spectrum for \( G \) similarly. Finally, we show that a tree is \( k \)-magic, \( k > 2 \), if and only if it is \( k \)-label reducible.
A graph \( G \) is 3-e.c. if for each distinct triple \( S \) of vertices, and each subset \( T \) of \( S \), there is a vertex not in \( S \) joined to the vertices of \( T \) and to no other vertices of \( S \). Few explicit examples of 3-e.c. graphs are known, although almost all graphs are 3-e.c. We provide new examples of 3-e.c. graphs arising as incidence graphs of partial planes resulting from affine planes. We also present a new graph operation that preserves the 3-e.c. property.
It is known that if a \( (22,33,12,8,4) \)-BIBD exists, then its incidence matrix is contained in a \( (33,16) \) doubly-even self-orthogonal code (that does not contain a coordinate of zeros). There are 594 such codes, up to equivalence. It has been theoretically proven that 116 of these codes cannot contain the incidence matrix of such a design. For the remaining 478 codes, an exhaustive clique search may be tried, on the weight 12 words of a code, to determine whether or not it contains such an incidence matrix. Thus far, such a search has been used to show 299 of the 478 remaining codes do not contain the incidence matrix of a \( (22,33,12,8,4) \)-BIBD.
In this paper, an outline of the method used to search the weight 12 words of these codes is given. The paper also gives estimations on the size of the search space for the remaining 179 codes. Special attention is paid to the toughest cases, namely the 11 codes that contain 0 weight 4 words and the 21 codes that contain one and only one weight 4 word.
Given a polyomino \( P \) with \( n \) cells, two players \( A \) and \( B \) alternately color the cells of the square tessellation of the plane. In the case of \( A \)-achievement, player \( A \) tries to achieve a copy of \( P \) in his color and player \( B \) tries to prevent \( A \) from achieving a copy of \( P \). The handicap number \( h(P) \) denotes the minimum number of cells such that a winning strategy exists for player \( A \). For all polyominoes that form a square of \( n = s^2 \) square cells, the handicap number will be determined to be \( s^2 – 1 \).
De Launey and Seberry have looked at the existence of Generalized Bhaskar Rao designs with block size 4 signed over elementary Abelian groups and shown that the necessary conditions for the existence of a \( (v, 4, \lambda; EA(g)) \) GBRD are sufficient for \( \lambda > g \) with 70 possible basic exceptions. This article extends that work by reducing those possible exceptions to just a \( (9, 4, 18h; EA(9h)) \) GBRD, where \( \gcd(6, h) = 1 \), and shows that for \( \lambda = g \) the necessary conditions are sufficient for \( v > 46 \).
Let \(G = (V,E)\) be an n-vertex graph and \(f : V \rightarrow \{1,2,\ldots,n\}\) be a bijection. The additive bandwidth of \(G\), denoted \(B^+(G)\), is given by \(B^+(G) = \min_{f} \max_{u,v\in E} |f(u) + f(v) – (n+1)|\), where the minimum ranges over all possible bijections \(f\). The additive bandwidth cannot decrease when an edge is added, but it can increase to a value which is as much as three times the original additive bandwidth. The actual increase depends on \(B^+(G)\) and n and is completely determined.
In Minimal Enclosings of Triple Systems I, we solved the problem of minimal enclosings of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for \(1 \leq \lambda \leq 6\) with a minimal \(m \geq 1\). Here we consider a new problem relating to the existence of enclosings for triple systems for any \(v\), with \(1 < 4 < 6\), of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+s, 3, \lambda+1)\) for minimal positive \(s\). The non-existence of enclosings for otherwise suitable parameters is proved, and for the first time the difficult cases for even \(\lambda\) are considered. We completely solve the case for \(\lambda \leq 3\) and \(\lambda = 5\), and partially complete the cases \(\lambda = 4\) and \(\lambda = 6\). In some cases a \(1\)-factorization of a complete graph or complete \(n\)-partite graph is used to obtain the minimal enclosing. A list of open cases for \(\lambda = 4\) and \(\lambda = 6\) is attached.
Halin’s Theorem characterizes those locally finite infinite graphs that embed in the plane without accumulation points by giving a set of six topologically-excluded subgraphs. We prove the analogous theorem for graphs that embed in an open Möbius strip without accumulation points. There are \(153\) such obstructions under the ray ordering defined herein. There are \(350\) obstructions under the minor ordering. There are \(1225\) obstructions under the topological ordering. The relationship between these graphs and the obstructions to embedding in the projective plane is similar to the relationship between Halin’s graphs and \(\{K_5, K_{3,3}\}.^1\)
In [5] Pila presented best possible sufficient conditions for a regular \(\sigma\)-connected graph to have a \(1\)-factor, extending a result of Wallis [7]. Here we present best possible sufficient conditions for a \(\sigma\)-connected regular graph to have a \(k\)-factor for any \(k \geq 2\).
We find a maximal number of directed circuits (directed cocircuits) in a base of a cycle (cut) space of a digraph. We show that this space has a base composed of directed circuits (directed cocircuits) if and only if the digraph is totally cyclic (acyclic). Furthermore, this basis can be considered as an ordered set so that each element of the basis has an arc not contained in the previous elements.
In this paper, we show that if \(G\) is a harmonious graph, then \((2n+1)G\) (the disjoint union of \(2n+1\) copies of \(G\)) and \(G ^{(2n+1)}\) (the graph consisting of \(2n+1\) copies of \(G\) with one fixed vertex in common) are harmonious for all \(n \geq 0\).
A critical set in a Latin square of order \(n\) is a set of entries from the square which can be embedded in precisely one Latin square of order \(n\), such that if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various “strengths”. Some observations are made about the relationship between the numbers of classes, particularly in the \(6 \times 6\) case. Finally some examples are given of each type of critical set.
A proper vertex \(k\)-coloring of a graph \(G\) is dynamic if for every vertex \(v\) with degree at least \(2\), the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a dynamic \(k\)-coloring is the dynamic chromatic number \(\chi_d(G)\). We prove in this paper the following best possible upper bounds as an analogue to Brook’s Theorem, together with the determination of chromatic numbers for complete \(k\)-partite graphs.
If \(x\) is a vertex of a digraph \(D\), then we denote by \(d^+(x)\) and \(d^-(x)\) the outdegree and the indegree of \(x\), respectively. The global irregularity of a digraph \(D\) is defined by \(i_g(D) = \max\{d^+(x),d^-(x)\} – \min\{d^+(y),d^-(y)\}\) over all vertices \(x\) and \(y\) of \(D\) (including \(x = y\)). If \(i_g(D) = 0\), then \(D\) is regular and if \(i_g(D) \leq 1\), then \(D\) is almost regular.
A \(c\)-partite tournament is an orientation of a complete \(c\)-partite graph. It is easy to see that there exist regular \(c\)-partite tournaments with arbitrarily large \(c\) which contain arcs that do not belong to a directed cycle of length \(3\). In this paper we show, however, that every arc of an almost regular \(c\)-partite tournament is contained in a directed cycle of length four, when \(c \geq 8\). Examples show that the condition \(c \geq 8\) is best possible.
We address the following problem: What minimum degree forces a graph on \(n\) vertices to have a cycle with at least \(c\) chords? We prove that any graph with minimum degree \(\delta\) has a cycle with at least \(\frac{(\delta+1)(\delta-2)}{2}\) chords. We investigate asymptotic behaviour for large \(n\) and \(c\) and we consider the special case where \(n = c\).
We prove that a finite set \(A\) of points in the \(n\)-dimensional Euclidean space \(\mathcal{R}^n\) is uniquely determined up to translation by three of its subsets of cardinality \(|A|-1\) given up to translation, i.e. the Reconstruction Number of such objects is three. This result is best-possible.
We solve the problem of existence of minimal enclosings for triple systems with \(1 \leq \lambda \leq 6\) and any \(v\), i.e., an inclusion of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for minimal positive \(m\). A new necessary general condition is derived and some general results are obtained for larger \(\lambda\) values.
Colour the edges of a \(K_{24n+1}\) by \(12\) colours so that every vertex in every colour has degree \(2n\). Is there a totally multicoloured \(C_4\) (i.e. every edge gets a different colour)? Here we answer in the affirmative to this question. In [1] P. Erdős stated the same problem for \(K_{12n+1}\) and \(6\) colours, it was settled in [2].
In this paper we follow the terminology and symbols of [3]. We assume the complete graph \(K_{24n+1}\) to have the vertex-set \(V=V(K_{24n+1}) = \{1, 2, \ldots, 24n+1\}\).
Let \(P(G)\) denote the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are chromatically equivalent, written \(G \sim H\), if \(P(G) = P(H)\). A graph \(G\) is chromatically unique if for any graph \(H\), \(G \sim H\) implies that \(G\) is isomorphic with \(H\). In “Chromatic Equivalence Classes of Certain Generalized Polygon Trees”, Discrete Mathematics Vol. \(172, 108–114 (1997)\), Peng \(et\; al\). studied the chromaticity of certain generalized polygon trees. In this paper, we present a chromaticity characterization of another big family of such graphs.
The step domination number of all graphs of diameter two is determined.
We use generator matrices \(G\) satisfying \(GG^T = aI + bJ\) over \(\mathbb{Z}_k\) to obtain linear self-orthogonal and self-dual codes. We give a new family of linear self-orthogonal codes over \(\text{GF}(3)\) and \(\mathbb{Z}_4\) and a new family of linear self-dual codes over \(\text{GF}(3)\).
Let \(S\) be a simply connected orthogonal polygon in the plane. Assume that the vertex set of \(S\) may be partitioned into sets \(A, B\) such that for every pair \(x, y\) in \(A\) (in \(B\)), \(S\) contains a staircase path from \(x\) to \(y\). Then \(S\) is a union of two or three orthogonally convex sets. If \(S\) is star-shaped via staircase paths, the number two is best, while the number three is best otherwise. Moreover, the simple connectedness requirement cannot be removed. An example shows that the segment visibility analogue of this result is false.
For a graph \(G\) of size \(m \geq 1\) and edge-induced subgraphs \(F\) and \(H\) of size \(r\) (\(1 \leq r \leq m\)), the subgraph \(Z\) is said to be obtained from \(F\) by an edge jump if there exist four distinct vertices \(u, v, w\), and \(x\) in \(G\) such that \(uv \in E(F)\), \(wx \in E(G) – E(F)\), and \(H = F – uv + wx\). The minimum number of edge jumps required to transform \(F\) into \(H\) is the jump distance from \(F\) to \(H\). For a graph \(G\) of size \(m \geq 1\) and an integer \(r\) with \(1 \leq r \leq m\), the \(r\)-jump graph \(J_r(G)\) is that graph whose vertices correspond to the edge-induced subgraphs of size \(r\) of \(G\) and where two vertices of \(J_r(G)\) are adjacent if and only if the jump distance between the corresponding subgraphs is \(1\). For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J_r(J^{k-1}_{r}(G))\), where \(J^1_r(G) = J_r(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. It is shown that if \(\{J^k_2(G)\}\) is a nonplanar sequence, then \(J^k_2(G)\) is nonplanar for all \(k \geq 3\) and there is only one graph \(G\) such that \(J^2_2(G)\) is planar. Moreover, for each integer \(r \geq 3\), if \(G\) is a connected graph of size at least \(r + 2\) for which \(\{J^k_r(G)\}\) is a nonplanar sequence, then \(J^k_r(G)\) is nonplanar for all \(k \geq 3\).
Let \(G\) be a finite group written additively and \(S\) a non-empty subset of \(G\). We say that \(S\) is \(e-exhaustive\) if \(G = S + \cdots + S\) (\(e\) times). The minimal integer \(e > 0\), if it exists, such that \(S\) is \(e-exhaustive\), is called the exhaustion number of the set \(S\) and is denoted by \(e(S)\). In this paper, we completely determine the exhaustion numbers of subsets of Abelian groups which are in arithmetic progression. The exhaustion numbers of various subsets of Abelian groups which are not in arithmetic progression are also determined.
Given graphs \(G\) and \(H\), an edge coloring of \(G\) is called an \((H,q)\)-coloring if the edges of every copy of \(H \subset G\) together receive at least \(q\) colors. Let \(r(G,H,q)\) denote the minimum number of colors in a \((H,q)\)-coloring of \(G\). In [6] Erdős and Gyárfás studied \(r(K_n,K_p,q)\) if \(p\) and \(q\) are fixed and \(n\) tends to infinity. They determined for every fixed \(p\) the smallest \(q\) for which \(r(K_n,K_p,q)\) is linear in \(n\) and the smallest \(q\) for which \(r(K_n,K_p,q)\) is quadratic in \(n\). In [9] we studied what happens between the linear and quadratic orders of magnitude. In [2] Axenovich, Füredi, and Mubayi generalized some of the results of [6] to \(r(K_{n,n},K_{p,p},q)\). In this paper, we adapt our results from [9] to the bipartite case, namely we study \(r(K_{n,n},K_p,p,q)\) between the linear and quadratic orders of magnitude. In particular, we show that we can have at most \(\log p + 1\) values of \(q\) which give a linear \(r(K_{n,n},K_{p,p},q)\).
In this paper, we define the concept of generalized Fibonacci polynomial of a graph \(G\) which gives the total number of all \(k\)-stable sets in generalized lexicographical products of graphs. This concept generalizes the Fibonacci polynomial of a graph introduced by G. Hopkins and W. Staton in [3].
A Fibonacci string of order \(n\) is a binary string of length \(n\) with no two consecutive ones. The Fibonacci cube \(\Gamma_n\) is the subgraph of the hypercube \(Q_n\) induced by the set of Fibonacci strings of order \(n\). For positive integers \(i, n\), with \(n \geq i\), the \(i\)th extended Fibonacci cube is the vertex-induced subgraph of \(Q_n\) for which \(V(\Gamma_{i}^{n}) = V_i\) is defined recursively by
\[V_{n+2}^{i} = 0 V_{n+1}^{i} + 10V_n^{i},\]
with initial conditions \(V_i^i = B_i, V_{i+1}^{i} = B_{i+1}\), where \(B_k\) denotes the set of binary strings of length \(k\). In this study, we answer in the affirmative a conjecture of Wu [10] that the sequences \(\{|V_n^i|\}_{i={1+2}}^\infty\) are pairwise disjoint for all \(i \geq 0\), where \(V_n^0 = V(\Gamma_n)\).
Let \(S\) be a simple polygon in the plane whose vertices may be partitioned into sets \(A’, B’\), such that for every two points of \(A’\) (of \(B’\)), the corresponding segment is in \(S\). Then \(S\) is a union of \(6\) (or possibly fewer) convex sets. The number \(6\) is best possible. Moreover, the simple connectedness requirement for set \(S\) cannot be removed.
An \(\lambda\)-design on \(v\) points is a set of \(v\) distinct subsets (blocks) of a \(v\)-element set (points) 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 all \(\lambda\)-designs can be obtained from symmetric designs by a certain complementation procedure. The main result of the present paper is that the \(\lambda\)-design conjecture is true when \(v = 8p + 1\), where \(p \equiv 1\) or \(7\) (mod \(8\)) is a prime number.
For an ordered set \(W = \{w_1, w_2, \ldots, w_e\}\) of vertices and a vertex \(v\) in a connected graph \(G\), the representation of \(v\) with respect to \(W\) is the \(e\)-vector \(r(v|W) = (d(v, w_1), d(v, w_2), \ldots, d(v, w_k))\), where \(d(x, y)\) represents the distance between the vertices \(x\) and \(y\). The set \(W\) is a resolving set for \(G\) if distinct vertices of \(G\) have distinct representations with respect to \(W\). A resolving set for \(G\) containing a minimum number of vertices is a basis for \(G\). The dimension \(\dim(G)\) is the number of vertices in a basis for \(G\). A resolving set \(W\) of \(G\) is connected if the subgraph \(\langle W \rangle\) induced by \(W\) is a connected subgraph of \(G\). The minimum cardinality of a connected resolving set in a graph \(G\) is its connected resolving number \(cr(G)\). The relationship between bases and minimum connected resolving sets in a graph is studied. A connected resolving set \(W\) of \(G\) is a minimal connected resolving set if no proper subset of \(W\) is a connected resolving set. The maximum cardinality of a minimal connected resolving set is the upper connected resolving number \(cr^+(G)\). The upper connected resolving numbers of some well-known graphs are determined. We present a characterization of nontrivial connected graphs of order \(n\) with upper connected resolving number \(n-1\). It is shown that for a pair \(a,b\) of integers with \(1 \leq a \leq b\) there exists a connected graph \(G\) with \(cr(G) = a\) and \(cr^+(G) = b\) if and only if \((a,b) \neq (1,4)\) for all \(i > 2\).
If \(G\) and \(H\) are graphs, define the Ramsey number \(r(G, H)\) to be the least number \(p\) such that if the edges of the complete graph \(K_p\) are colored red and blue (say), either the red graph contains a copy of \(G\), or the blue graph contains a copy of \(H\). In this paper, we determine the Ramsey number \(r(mC_4, nC_5)\) for any \(m\geq1, n\geq1\).
We construct a complex \(K^n\) of \(m\)-ary relations, \(1 \leq m \leq n+1\), in a finite set \(X \neq Ø\), representing a model of an abstract cellular complex. For such a complex \(K^n\) we define the matrices of incidence and coincidence, the groups of homologies \(\mathcal{H}_m(K^n)\) and cohomologies \(\mathcal{H}^m(K^n)\) on the group of integers \(\mathbf{Z}\), and the Euler characteristic. On a combinatorial basis we derive their main properties. In further publications we will derive more analogues of classical properties, and also applications with respect to the existence of fixed relations in the utilization of the isomorphisms will be investigated. In particular, we intend to complete the theory of hypergraphs with the help of such topological observations.
Colbourn introduced \(V_\lambda(m, t)\) to construct transversal designs with index \(\lambda\). A \(V_\lambda(m, t)\) leads to a \((mt + 1, mt + 2; \lambda,0; t)\)-aussie-difference matrix. In this article, we use Weil’s theorem on character sums to show that for any integer \(\lambda \geq 2\), a \(V_\lambda(m, t)\) always exists in \(GF(mt + 1)\) for any prime power \(mt+1 > B_\lambda(m) = \left[\frac{E+\sqrt{E^2+4F}}{2}\right]^2\), where \(E = \lambda(u-1)(m-1)m^u-m^{u-1}+1,F=(u-1)\lambda m^u\) and \(u = \left\lfloor\frac{m{\lambda}+1+(-1)^{\lambda+1}}{2}\right\rfloor\). In particular, we determine the existence of \(V_{\lambda}(m, t)\) for \((\lambda, m) = (2, 2), (2, 3)\).
A weighted graph \((G,w)\) is a graph \(G = (V, E)\) together with a positive weight-function on its vertices \(w: V \to \mathbf{R}^{>0}\). The weighted domination number \(\gamma_w(G)\) of \((G, w)\) is the minimum weight \(w(D) = \sum_{v \in D} w(v)\) of a vertex set \(D \subseteq V\) with \(N[D] = V\), i.e. a dominating set of \(G\).
For this natural generalization of the well-known domination number, we study some of the classical questions of domination theory. We characterize all extremal graphs for the simple Ore-like bound \(\gamma_w(G) \leq \frac{1}{2}w(V)\) and prove Nordhaus-Gaddum-type inequalities for the weighted domination number.
Generalized Steiner systems GS\(_d(t,k,v,g)\) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size \(g + 1\) with minimum Hamming distance \(d\), in which each codeword has length \(v\) and weight \(k\). It was proved that the necessary conditions for the existence of a GS\(_4(2,4,v,g)\) are also sufficient for \(g = 2, 3\) and \(6\). In this paper, a general result on the existence of a GS\(_4(2,4,v,g)\) is presented. By using this result, we prove that the necessary conditions \(v \equiv 1 \pmod{3}\) and \(v \geq 7\) are also sufficient for the existence of a GS\(_4(2, 4, v, 4)\).
An \(A^3\)-code is an extension of an \(A\)-code in which none of the three participants, transmitter, receiver, and arbiter, is trusted. In this paper, we extend the previous model of \(A^3\)-codes by allowing the transmitter and the receiver to not only individually attack the system, but also collude with the arbiter against the other. We derive information-theoretic lower bounds on the success probability of various attacks, and combinatorial lower bounds on the size of key spaces. We also study the combinatorial structure of optimal \(A^3\)-codes against collusion attacks and give a construction of an optimal code.
We prove that \(15\) is the maximal size of a \(3\)-arc in the projective plane of order \(8\).
A \(k\)-line-distinguishing coloring of a graph \(G = (V,E)\) is a partition of \(V\) into \(k\) sets \(V_1, V_2, \ldots, V_k\) such that \(q(\langle V_i \rangle) \leq 1\) for \(i = 1, \ldots, k\) and \(q(V_i . V_j) \leq 1\) for \(1 \leq i < j \leq k\). If the color classes in a line-distinguishing coloring are also independent. Then it is called a harmonious coloring. A coloring is minimal if when two color classes are combined, we no longer have a coloring of the given type. The upper harmonious chromatic number, \(H(G)\), is defined as the maximum cardinality of a minimal harmonious coloring of a graph \(G\). While the upper line-distinguishing chromatic number, \(H'(G)\), is defined as the maximum cardinality of a minimal line-distinguishing coloring of a graph \(G\). We determine \(H'(C_n)\) and \(H(C_n)\) for an even cycle \(C_n\).
In this paper, we derive an inequality on the existence of bi-level balanced arrays (B-arrays) of strength eight by using a result involving central moments from statistics, and by counting in two ways the number of coincidences of various columns with a specific column. We discuss the use of this inequality in obtaining the maximum number of constraints for these arrays, and present some illustrative examples.
In this note, we present many uniquely \(n\)-colorable graphs with \(m\) vertices and new constructing ways of uniquely colorable graphs by using the theory of adjoint polynomials of graphs. We give new constructing ways of two uniquely colorable graphs which are chromatically equivalent also.
An \(LD(n,k,p,t:b)\) Lotto design is a set of \(b\) \(k\)-sets (blocks) of an \(n\)-set such that any \(p\)-set intersects at least one block in \(t\) or more elements. Let \({L}(n,k,p,t)\) denote the minimum number of blocks for any \(LD(n,k,p,t:b)\) Lotto design. This paper describes an algorithm used to construct Lotto designs by combining genetic algorithms and simulated annealing and provides some experimental results.
A union closed (UC) family \(\mathcal{A}\) is a finite family of sets such that the union of any two sets in \(\mathcal{A}\) is also in \(\mathcal{A}\). Peter Frankl conjectured that for every union closed family \(\mathcal{A}\), there exists some \(x\) contained in at least half the members of \(\mathcal{A}\). This is the union-closed sets conjecture.
An FC family is a UC family \(\mathcal{B}\) such that for every UC family \(\mathcal{A}\), if \(\mathcal{B} \subseteq \mathcal{A}\), then \(\mathcal{A}\) satisfies the union-closed sets conjecture. We give a heuristic method for identifying possible FC families, and apply it to families in \(\mathcal{P}(5)\) and \(\mathcal{P}(6)\).
The vertices \(V\) of trees with maximum degree three and \(t\) degree two vertices are partitioned into sets \(R\), \(B\), and \(U\) such that the induced subgraphs \(\langle V – R \rangle\) and \(\langle V – B \rangle\) are isomorphic and \(|U|\) is minimum. It is shown for \(t \geq 2\) that there is such a partition for which \(|U| = 0\) if \(t\) is even and \(|U| = 1\) if \(t\) is odd. This extends earlier work by the authors which answered this problem when \(t = 0\) or \(1\).
Let G be a simple connected graph on 2n vertices with a perfect matching. For a positive integer k, \(1 \leq \text{k} \leq \text{n}-1\), G is \(k\)-extendable if for every matching M of size k in G, there is a perfect matching in G containing all the edges of M. For an integer k, \(0 \leq \text{k} \leq \text{n} – 2\), G is trongly \(k\)-extendable if \(\text{G} – \{\text{u, v}\}\) is \(k\)-extendable for every pair of vertices u and v of G. The problem that arises is that of characterizing k-extendable graphs and strongly k-extendable graphs. The first of these problems has been considered by several authors while the latter has been recently investigated. In this paper, we focus on a minimum cutset of strongly k-extendable graphs. For a minimum cutset S of a strongly k-extendable graph G, we establish that if \(|\text{S}| = \text{k + t}\), for an integer \(\text{t} \geq 3\), then the independence number of the induced subgraph G[S] is at most \(2\) or at least k + 5 – t. Further, we present an upper bound on the number of components of G – S.
Let \(\{G_{pn} | n \geq 1\} = \{G_{p1}, G_{p2}, G_{p3}, \ldots\}\) be a countable sequence of simple graphs, where \(G_{pn}\) has \(pn\) vertices. This sequence is called \(K_p\)-removable if \(G_{p1} = K_p\), and \(G_{pn} – K_p = G_{p(n-1)}\) for every \(n \geq 2\) and for every \(K_p\) in \(G_{pn}\). We give a general construction of such sequences. We specialize to sequences in which each \(G_{pn}\) is regular; these are called regular \((K_p, \lambda)\)-removable sequences, where \(\lambda\) is a fixed number, \(0 \leq \lambda \leq p\), referring to the fact that \(G_{pn}\) is \((\lambda(n – 1) + p – 1)\)-regular. We classify regular \((K_p, 0)\)-, \((K_p, p – 1)\)-, and \((K_p, p)\)-removable sequences as the sequences \(\{nK_p | n \geq 1\}\), \(\{K_{p \times n} | n \geq 1\}\), and \(\{K_{pn} | n \geq 1\}\) respectively. Regular sequences are also constructed using `levelled’ Cayley graphs, based on a finite group. Some examples are given.
A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(u, v,\) and \(w\) with \(d(u,v) = 2\) and \(w \in N(u) \cap N(v)\), \(d(u) + d(v) \geq |N(u) \cup N(v) \cup N(w)| – 1\). Let \(G\) be a 2-connected \(L_1\)-graph of order \(n\). If \(\sigma_3(G) \geq n – 2\), then \(G\) is hamiltonian or \(G \in \mathcal{K}\), where \(\sigma_3(G) = \min\{d(u) + d(v) + d(w) : \{u,v,w\} \text{ is an independent set in } G\}\), \(\mathcal{K}=\{G: K_{p, p+1} \subseteq G \subseteq K_p + (p+1)K_1 for some p \geq 2\}\). A similar result on the traceability of connected \(L_1\)-graphs is also obtained.
For a graph \(G\), the jump graph \(J(G)\) is that graph whose vertices are the edges of \(G\) and where two vertices of \(J(G)\) are adjacent if the corresponding edges are not adjacent. For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J(J^{k-1}(G))\), where \(J^1(G) = J(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. All connected graphs \(G\) for which \(\{J^k(G)\}\) is planar have been determined. In this paper, we investigate those connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar. It is shown that if \(\{J^k(G)\}\) is a nonplanar sequence, then \(J^k(G)\) is nonplanar for all \(k \geq 4\). Furthermore, there are only six connected graphs \(G\) for which \(\{J^k(G)\}\) is nonplanar and \(J^3(G)\) is planar.
We examine a query posed as a conjecture by Key and Moori [11, Section 7] concerning the full automorphism groups of designs and codes arising from primitive permutation representations of finite simple groups, and based on results for the Janko groups \(J_1\) and \(J_2\) as studied in [11]. Here, following that same method of construction, we show that counter-examples to the conjecture exist amongst some representations of some alternating groups, and that the simple symplectic groups in their natural representation provide an infinite class of counter-examples.
We give six improved bounds on \(A(n,d,w)\), the maximum cardinality of a binary code of length \(n\) with minimum distance \(d\) and constant weight \(w\).
In this paper, we show that if \(G\) is an “\(\alpha\)-labeled” graph and if \(H\) is a “pseudograceful” graph, then \(G \cup H\) can be graceful or “pseudograceful” under some conditions on the \(\alpha\)-labeling function of \(G\). This generalizes Theorem 2.1 of [21]. We also show that if \(G\) is a Skolem-graceful, then \(G + \overline{K_n}\) is graceful for all \(n \geq 1\). We also give a partial answer to the question in [1] about the gracefulness of \(\overline{K_n} + mK_2\) for \(m \geq 3\). Finally, we complete the characterization of graceful graphs in the family \(C_m \cup S_n\).
We study the discrete version of the \(p\)-Laplacian operator — \(\textrm{div}(|\nabla u|^{p-2}\nabla u)\) — and we give some estimates of its smallest positive eigenvalue. In earlier papers, eigenvalues of the discrete Laplacian have been considered. We shall here study more general means. We shall also, in particular, study the case when the graph is complete. We give an estimate of the smallest positive eigenvalue of the \(p\)-Laplacian when the graph is a subgraph of \(\mathbb{Z}^n\) in this context. We give all eigenvalues of the \(p\)-Laplacian when the graph is complete.
Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as AT-free graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipartite permutation graphs by showing that any connected graph in the class can be stretched into a “path” with “edges” being chain graphs. A particular consequence from the obtained characterization is that the clique-width of bipartite permutation graphs is unbounded, which refines a recent result of Golumbic and Rotics for permutation graphs.
A known result due to Matthews and Sunner is that every \(2\)-connected claw-free graph on \(n\) vertices contains a cycle of length at least \(\min\{2\delta+4,n\}\), and is Hamiltonian if \(n \leq 3\delta+2\). In this paper, we show that every \(2\)-connected claw-free graph on \(n\) vertices which does not belong to one of three classes of exceptional graphs contains a cycle of length at least \(\min\{4\delta-2,n\}\), hereby generalizing several known results. Moreover, the bound \(4\delta-2\) is almost best possible.
A graph or a digraph \(G\) 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 \(G\) is super-\(\lambda\), then \(\lambda(G) = \delta(G)\), where \(\delta(G)\) is the minimum degree and \(\lambda(G)\) is the edge-connectivity of \(G\).
In this paper, degree sequence conditions for graphs and digraphs as well as for bipartite graphs and digraphs to be super-\(\lambda\) are presented.
Given integers \(k \geq 2\) and \(n \geq k\), let \(e(n, k)\) denote the maximum possible number of edges in an \(m\)-vertex graph which has no \(k\)-connected subgraph. It is immediate that \(e(n, 2) = n – 1\). Mader [2] conjectured that for every \(k \leq 2\), if \(n\) is sufficiently large then \(c(n, k) \leq (1.5k-2)(n – k + 1),\) where equality holds whenever \(k – 1\) divides \(n\). In this note we prove that when \(n\) is sufficiently large then \(e(n, k) \leq \frac{193}{120}(k – 1)(n – k + 1) < 1.61(k – 1)(n – k + 1),\) thereby coming rather close to the conjectured bound.
In this paper, we give a few applications of combinatorial design theory to a few problems in extremal graph theory. Using known results in combinatorial design theory, we have unified, simplified, and extended results on a few problems.
Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). A vertex labeling \(\overline{f}: V \to \{0,1\}\) induces an edge labeling \(\overline{f}: E \to \{0,1\}\) defined by \(f(uv) = |f(u) – f(v)|\). Let \(v_f(0),v_f(1)\) denote the number of vertices \(v\) with \(f(v) = 0\) and \(f(v) = 1\) respectively. Let \(e_f(0),e_f(1)\) be similarly defined. A graph is said to be cordial if there exists a vertex labeling \(f\) such that \(|v_f(0) – vf(1)| \leq 1\) and \(|e_f(0) – e_f(1)| \leq 1\).
A \(t\)-uniform homeomorph \(P_t(G)\) of \(G\) is the graph obtained by replacing all edges of \(G\) by vertex disjoint paths of length \(t\). In this paper we show that (1)\(P_t(K_{2n})\) is cordial for all \(t \geq 2\).(2) \(P_t(K_{2n+1})\) is cordial if and only iff (a) \(t \equiv 0 \pmod{4}\), or(b) \(t\) is odd and \(n\) is not \(\equiv 2 \pmod{4}\), or (c) \(t \equiv 2 \pmod{4}\) and \(n\) is even.
For any positive integer \(k\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_k\)-magic if there exists a labeling \(l: E(G) \to \mathbb{Z}_k – \{0\}\) such that the induced vertex set labeling \(l^+: V(G) \to \mathbb{Z}_k\) defined by
\[l^+(v) = \sum\{l(uv): uv \in E(G)\}\]
is a constant map. For a given graph \(G\), the set of all \(h \in \mathbb{Z_+}\) for which \(G\) is \(\mathbb{Z}_h\)-magic is called the integer-magic spectrum of \(G\) and is denoted by \(IM(G)\). In this paper, we will determine the integer-magic spectra of the graphs which are formed by the amalgamation of stars and cycles. In particular, we will provide examples of graphs that for a given \(n > 2\), they are not \(h\)-magic for all values of \(2 \leq k \leq n\).
In this paper, various transformations of the set of closed meanders are introduced. Some of these are used in order to partition the above set and to find a representative of each class. Furthermore, each closed meander is separated into shorter ones.
We develop a combinatorial model of paperfolding for the purposes of enumeration. A planar embedding of a graph is called a crease pattern if it represents the crease lines needed to fold a piece of paper into something. A flat fold is a crease pattern which lies flat when folded, i.e., can be pressed in a book without crumpling. Given a crease pattern \(C = (V, E)\), a mountain-valley (MV) assignment is a function \(f : E \to \{M, V\}\) which indicates which crease lines are convex and which are concave, respectively. A MV assignment is valid if it doesn’t force the paper to self-intersect when folded. We examine the problem of counting the number of valid MV assignments for a given crease pattern. In particular, we develop recursive functions that count the number of valid MV assignments for flat vertex folds, crease patterns with only one vertex in the interior of the paper. We also provide examples, especially those of Justin, that illustrate the difficulty of the general multivertex case.
An asteroidal triple is an independent set of three vertices in a graph such that every two of them are joined by a path avoiding the closed neighborhood of the third. Graphs without asteroidal triples are called AT-free graphs. In this paper, we show that every AT-free graph admits a vertex ordering that we call a \(2\)-cocomparability ordering. The new suggested ordering generalizes the cocomparability ordering achievable for cocomparability graphs. According to the property of this ordering, we show that every proper power \(G^k\) (\(k \geq 2\)) of an AT-free graph \(G\) is a cocomparability graph. Moreover, we demonstrate that our results can be exploited for algorithmic purposes on AT-free graphs.
We exhibit some problems definable in Feder and Vardi’s logic \(MMSNP\) that are not in the class \(CSP\) of constraint satisfaction problems. Whilst some of these problems have previously been shown to be in \(MMSNP\) (that is, definable in \(MMSNP\)) but not in \(CSP\), existing proofs are probabilistic in nature. We provide explicit combinatorial constructions to prove that these problems are not in \(CSP\) and we use these constructions to exhibit yet more problems in \(MMSNP\) that are not in \(CSP\).
The distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining \(u\) and \(v\). The eccentricity \(e(v)\) of vertex \(v\) is the distance to a vertex farthest from \(v\). In a graph \(G\), an eccentric vertex of \(v\) is a vertex farthest from \(v\), that is, a vertex \(u\) for which \(d(u,v) = e(v)\). Given a set \(X\) of vertices in \(G\), the vertices of \(X\) are mutually eccentric provided that for any pair of vertices \(u\) and \(v\) in \(X\), \(u\) is an eccentric vertex of \(v\) and \(v\) is an eccentric vertex of \(u\). In this paper, we discuss problems concerning sets of mutually eccentric vertices in graphs.
A \(k\)-circular-distance-two labeling (or \(k\)-c-labeling) of a simple graph \(G\) is a vertex-labeling, using the labels \(0, 1, 2, \ldots, k-1\), such that the “circular difference” (mod \(k\)) of the labels for adjacent vertices is at least two, and for vertices of distance-two apart is at least one. The \(\sigma\)-number, \(\sigma(G)\), of a graph \(G\) is the minimum \(k\) of a \(k\)-c-labeling of \(G\). For any given positive integers \(n\) and \(k\), let \(\mathcal {G}^{\sigma}(n, k)\) denote the set of graphs \(G\) on \(n\) vertices and \(\sigma(G) = k\). We determine the maximum size (number of edges) and the minimum size of a graph \(G \in \mathcal {G}^{\sigma}(n, k)\). Furthermore, we prove that for any value \(p\) between the maximum and the minimum size, there exists a graph \(G \in \mathcal {G}^{\sigma}(n, k)\) of size \(p\). These results are analogues of the ones by Georges and Mauro [4] on distance-two labelings.
We give a parametric representation for generic magic squares. This makes it relatively easy to construct magic squares having desired properties. It also suggests a convenient method for generating and classifying all the magic squares of every given order.
A vertex \(v\) in a digraph \(D\) out-dominates itself as well as all vertices \(u\) such that \((v,u)\) is an arc of \(D\); while \(v\) in-dominates both itself and all vertices \(w\) such that \((w,v)\) is an arc of \(D\). A set \(S\) of vertices of \(D\) is a twin dominating set of \(D\) if every vertex of \(D\) is out-dominated by some vertex of \(S\) and in-dominated by some vertex of \(S\). The minimum cardinality of a twin dominating set is the twin domination number \(\gamma^*(D)\) of \(D\). It is shown that \(\gamma^*(D) \leq \frac{2p}{3}\) for every digraph \(D\) of order \(p\) having no vertex of in-degree \(0\) or out-degree \(0\). Moreover, we give a Nordhaus-Gaddum type bound for \(\gamma^*\), and for transitive digraphs we give a sharp upper bound for the twin domination number in terms of order and minimum degree.
For a graph \(G\), the upper orientable twin domination number \(DOM^*(G)\) is the maximum twin domination number \(\gamma^*(D)\) over all orientations \(D\) of \(G\); while the lower orientable twin domination number \(dom^*(G)\) of \(G\) is the minimum such twin domination number. It is shown that for each graph \(G\) and integer \(c\) with \(dom^*(G) \leq c \leq DOM^*(G)\), there exists an orientation \(D\) of \(G\) such that \(\gamma^*(D) = c\).
For positive integers \(k \leq n\), the crown \(C_{n,k}\) is the graph with vertex set \(\{a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n\}\) and edge set \(\{a_ib_j: 1 \leq i \leq n, j = i,i+1,\ldots, i+k-1 \pmod{n}\}\). In this paper, we give a necessary and sufficient condition for the existence of a \(P_1\) decomposition of \(C_{n,k}\).
We use an array given in H. Kharaghani, “Arrays for orthogonal designs”, J. Combin. Designs, \(8 (2000), 166-173\), to obtain infinite families of \(8\)-variable Kharaghani type orthogonal designs, \(OD(8t; k_1, k_1, k_1, k_1, k_2, k_2, k_2, k_2)\), where \(k_1\) and \(k_2\) must be the sum of two squares. In particular, we obtain infinite families of \(8\)-variable Kharaghani type orthogonal designs, \(OD(8t; k, k, k, k, k, k, k, k)\). For odd \(t\), orthogonal designs of order \(\equiv 8 \pmod{16}\) can have at most eight variables.
We introduce semi quadrangles, which are finite partial linear spaces with a constant number of points on each line, having no ordinary triangles and containing, as minimal circuits, ordinary quadrangles and pentagons, with the additional property that every two non-collinear points are collinear with at least one other point of the geometry. A semi quadrangle is called thick if every point is incident with at least three lines and if every line is incident with at least three points. Thick semi quadrangles generalize (thick) partial quadrangles (see [4]). We will emphasize the special situation of the semi quadrangles which are subgeometries of finite generalized quadrangles. Some particular geometries arise in a natural way in the theory of symmetries of finite generalized quadrangles and in the theory of translation generalized quadrangles, as certain subgeometries of generalized quadrangles with concurrent axes of symmetry; these subgeometries have interesting automorphism groups, see [17] and also [19]. Semi quadrangles axiomatize these geometries. We will present several examples of semi quadrangles, most of them arising from generalized quadrangles or partial quadrangles. We will prove an inequality for semi quadrangles which generalizes the inequality of Cameron [4] for partial quadrangles, and the inequality of Higman [7,8] for generalized quadrangles. The proof also gives information about the equality. Some other inequalities and divisibility conditions are computed. Also, we will characterize the linear representations of the semi quadrangles, and we will have a look at the point graphs of semi quadrangles.
Let \(G\) be a graph, \(\overline{G}\) its complement, \(L(G)\) its line graph, and \(\chi(G)\) its chromatic number. Then we have the following
THEOREM Let \(G\) be a graph with \(n\) vertices. (i) If \(G\) is triangle
free, then
\[n-4 \leq \chi\left(\overline{L(\overline{G})}\right)\leq n-2\]
(ii) If G is planar and every triangle bounds a disk, then
\[n-3 \leq \chi\left(\overline{L(\overline{G})}\right)\leq n-2\]
Let \(G\) be a simple graph on \(n\) vertices with list chromatic number \(\chi_\ell = s\). If each vertex of \(G\) is assigned a list of \(t\) colours, Albertson, Grossman, and Haas [1] asked how many of the vertices, \(\lambda_{t,s}\), are necessarily colourable from these lists? They conjectured that \(\lambda_{t,s} \geq \frac{tn}{s}\). Their work was extended by Chappell [2]. We improve the known lower bounds for \(\lambda_{t,s}\).
In general, the class of threshold hypergraphs and decomposable hypergraphs are not equal. In this paper, we show however that, except for two counter examples, a decomposition hypergraph consisting of five or fewer classes is in fact threshold. In the process of showing this result, the paper generates all decomposable quotients with five or fewer classes.
In this paper, we show that for every sufficiently large integer \(n\) and every positive integer \(c \leq \left\lfloor \frac{1}{6}({\log \log n})^\frac{1}{2} \right \rfloor\), a Boolean lattice with \(n\) atoms can be partitioned into chains of cardinality \(c\), except for at most \(c-1\) elements which also form a chain.
We construct all self-dual \([24, 12, 8]\) quaternary codes with a monomial automorphism of prime order \(r > 3\) and obtain a unique code for \(r = 23\) (which has automorphisms of orders \(5\), \(7\), and \(11\) too), two inequivalent codes for \(r = 11\), \(6\) inequivalent codes for \(r = 7\), and \(12\) inequivalent codes for \(r = 5\). The obtained codes have \(12\) different weight spectra.
Metamorphoses of small \(k\)-wheel systems for \(k = 3, 4,\) and \(6\) are obtained. In particular, we obtain simultaneous metamorphoses of: \(3\)-wheel systems into Steiner triple systems and into \(K_{1,3}\)-designs; \(4\)-wheel systems into \(4\)-cycle systems, \(K_{1,4}\)-designs, and bowtie systems; \(6\)-wheel systems into \(6\)-cycle systems, \(K_{1,6}\)-designs, and \(3\)-windmill designs or near-\(3\)-windmill designs.
We deal with the problem of labeling the vertices, edges, and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all the faces constitute an arithmetical progression of difference \(d\).
If \(L\) is a list assignment function and \(\kappa\) is a multiplicity function on the vertices of a graph \(G\), a certain condition on \((G, L, \kappa)\), known as Hall’s multicoloring condition, is obviously necessary for the existence of a multicoloring of the vertices of \(G\). A graph \(G\) is said to be in the class \(MHC\) if it has a multicoloring for any functions \(L\) and \(\kappa\) such that \((G, L, \kappa)\) satisfies Hall’s multicoloring condition. It is known that if \(G\) is in \(MHC\) then each block of \(G\) is a clique and each cutpoint lies in precisely two blocks. We conjecture that the converse is true as well. It is also known that if \(G\) is a graph consisting of two cliques joined at a point then \(G\) is in \(MHC\). We present a new proof of this result which uses common partial systems of distinct representatives, the relationship between matching number and vertex covering number for 3-partite hypergraphs, and Menger’s Theorem.
This paper presents a new approach in the quest for a solution to the \(3x+1\) problem. The method relies on the convergence of the trajectories of the odd positive integers by exploiting the role of the positive integers of the form \(1+4n\), where \(n\) is a non-negative integer.
A cyclic or bicyclic \(9 \times 37\) double Youden rectangle (DYR) is provided for each of the four biplanes with \(k = 9\). These DYRs were obtained by computer search.
For loopless plane multigraphs \(G\), the edge-face chromatic number and the entire chromatic number are asymptotically their fractional counterparts (LP relaxations) as these latter invariants tend to infinity. Proofs of these results are based on analogous theorems for the chromatic index and the total chromatic number, due, respectively, to Kahn [3] and to the first author [6]. Our two results fill in the missing pieces of a complete answer to the natural question: which of the seven invariants associated with colouring the nonempty subsets of \(\{V, E, F\}\) exhibit “asymptotically good” behaviour?
The bin packing problem has been studied extensively since the 1970’s, and it is known to be applicable to many different areas, especially in operations research and computer science. In this paper, we present a variant of the classical bin packing problem, which allows the packing to exceed its bin size but at least a fraction of the last piece is within the bin, and we call it the open-end bin packing problem. This paper is focused on on-line open-end bin packing. An on-line open-end bin packing algorithm is to assign incoming pieces into the bins on-line, that is, there is no information about the sizes of the pieces in future arrivals. An on-line algorithm is optimal if it always produces a solution with the minimum number of bins used for packing. We show that no such optimal algorithm exists. We also present seven efficient on-line algorithms: Next Fit, Random Fit, Worst Fit, Best Fit, Refined Random Fit, Refined Worst Fit, and Refined Best Fit, which give sub-optimal solutions. The performances of these algorithms are studied. A case study for the application of the studied problem is presented, and this is a practical problem on maximizing the savings of using stored-value tickets issued by Kowloon-Canton Railway (KCR), which is one of the major public transportation means in Hong Kong.
We explore the maximum possible toughness among graphs with \(n\) vertices and \(m\) edges in the cases in which \(\lceil \frac{3n}{2}\rceil \leq m < 2n\). In these cases, it is shown that the maximum toughness lies in the interval \([\frac{4}{3}, \frac{3}{2}]\). Moreover, if \(\left\lceil\frac{3n}{2}\right\rceil + 2 \leq m < 2n\), then the value \(\frac{3}{2}\) is achieved. However, if \(m \in \left\{\left\lceil\frac{3n}{2}\right\rceil, \left\lceil\frac{3n}{2}\right\rceil + 1\right\}\), then the maximum toughness can be strictly less than \(\frac{3}{2}\). This provides an infinite family of graphs for which the maximum toughness is not half of the maximum connectivity. The values of maximum toughness are computed for all \(1 \leq n \leq 12\), and some open problems are presented.
A set \(S\) of vertices of a graph \(G = (V, E)\) is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). We define the total domination subdivision number \(sd_{\gamma t}(G)\) to be the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. We give upper bounds on the total domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on \(G\) sufficient to imply that \(sd_{\gamma t}(G) \leq 3\). On the other hand, we show that this constant upper bound does not hold for all graphs. Finally, we show that \(1 \leq sd_{\gamma t}(T) \leq 3\) for any tree \(T\), and characterize the caterpillars \(T\) for which \(sd_{\gamma t}(T) = 3\).
We show that for every \(d \geq 2\), the number of spanning trees of a \(d\)-dimensional grid with \(N\) vertices grows like \(C(d)^N\) for some constant \(C(d)\). Moreover, we show that \(C(d) = 2d-\frac{1}{2}-\frac{5}{16d} + O(d^{-2})\) as \(d\) goes to infinity.
An extended 5-cycle system of order \(n\) is an ordered pair \((V, B)\), where \(B\) is a collection of edge-disjoint 5-cycles, 2-tadpoles, and loops that partition the edges of the graph \(K_n^+\) whose vertex set is an \(n\)-set \(V\). In this paper, we show that an extended 5-cycle system of order \(n\) exists for all \(n\) except \(n = 2\) and \(3\).
McMorris, Zaslavsky, and Diny give characterizations of upper bound graphs and double bound graphs in terms of edge clique covers, that is, a family of maximal complete subgraphs that covers all edges. Lundgren and Maybee give a characterization of upper bound graphs using a concept of non-maximal complete subgraphs. In this paper, we present characterizations of double bound graphs and semi-bound graphs in terms of edge covers of non-maximal complete subgraphs.
We consider families of linear self-orthogonal and self-dual codes over the ring \({Z}_4\), which are generated by weighing matrices \(W(n, k)\) with \(k \equiv 0 \pmod{4}\), whose entries are interpreted as elements of the ring \({Z}_4\). We obtain binary formally self-dual codes of minimal Hamming distance 4 by applying the Gray map to the quaternary codes generated by \(W(n, 4)\).
Let \(G = (V, E)\) be a simple, undirected graph. A set of vertices \(D\) is called an odd dominating set if for every vertex \(v \in V(G)\), \(|N[v] \cap D| \equiv 1 \pmod{2}\). The minimum cardinality of an odd dominating set is called the odd domination number of \(G\). It is well known that every graph contains an odd dominating set, but this parameter has been studied very little. Our aim in this paper is to explore some basic features of the odd domination number and to compare it with the domination number of the graph, denoted by \(\gamma(G)\). In addition, extremal values of \(\gamma_{odd}(G)\) are calculated for several classes of graphs and a Nordhaus-Gaddum type inequality \(\gamma_{odd}(G) + \gamma_{odd}(\overline{G})\) is considered.
In this paper, it will be shown that a Skolem sequence of order \(n \equiv 0,1 \pmod{4}\) implies the existence of a graceful tree on \(2n\) vertices which exhibits a perfect matching or a matching on \(2n-2\) vertices. It will also be shown that a Hooked-Skolem sequence of order \(n \equiv 2,3 \pmod{4}\) implies the existence of a graceful tree on \(2n+1\) vertices which exhibits a matching on either \(2n\) or \(2n-2\) vertices. These results will be established using an algorithmic approach.
For \(k \geq 1\) an integer, a set \(D\) of vertices of a graph \(G = (V, E)\) is a \(k\)-dominating set of \(G\) if every vertex in \(V – D\) is within distance \(k\) from some vertex of \(D\). The \(k\)-domination number \(\gamma_k(G)\) of \(G\) is the minimum cardinality among all \(k\)-dominating sets of \(G\). For \(\ell \geq 2\) an integer, the graph \(G\) is \((\gamma_k, \ell)\)-critical if \(\gamma_k(G) = \ell\) and \(\gamma_k(G – v) = \ell – 1\) for all vertices \(v\) of \(G\). If \(G\) is \((\gamma_k, \ell)\)-critical for some \(\ell\), then \(G\) is also called a \(\gamma_k\)-critical graph. For a vertex \(v\) of \(G\), let \(N_k(v) = \{u \in V – \{v\} | d(u,v) \leq k\}\) and let \(\delta_k(G) = \min\{|N_k(v)|: v \in V\}\) and let \(\Delta_k(G) = \max\{|N_k(v)|: v \in V\}\). It is shown that if \(G\) is a nontrivial connected \(\gamma_k\)-critical graph, then \(\delta_k(G) \geq 2k\). Further, it is established that the number of vertices in a \(\gamma-k\)-critical graph \(G\) is bounded above by \((\Delta_k(G)+1)(\gamma_k(G)-1)+1\) and that \(G\) is a \((\gamma_k, \ell)\)-critical graph if and only if the \(k\)th power of \(G\) is a \((\gamma, \ell)\)-critical graph. It is shown that \((k, \ell)\)-critical graphs of arbitrarily large connectivity exist. Moreover, a graph without isolated vertices is shown to be \(\gamma_k\)-critical if and only if each of its blocks is \(\gamma_k\)-critical. Finally it is established that for an integer \(\ell \geq 2\), every graph is an induced subgraph of some \((\gamma_k, \ell)\)-critical graph. This paper concludes with some partially answered questions and some open problems.
We provide complete lists of starters and Skolem sequences which generate perfect one-factorizations of complete graphs up to order \(32\) for starters and \(36\) for Skolem sequences. The resulting perfect one-factorizations are grouped into isomorphism classes, and further analysis of the results is performed.
We find new full orthogonal designs in order 72 and show that of 2700 possible \(OD(72; s_1, s_2, s_3, 72 – s_1 – s_2 – s_3)\), 335 are known, of 432 possible \(OD(72; s_1, s_2, 72 – s_1 – s_2)\), 308 are known. All possible \(OD(72; s_1, 72 – s_1)\) are known.
Classical bin packing has been studied extensively in the literature. Open-ends bin packing is a variant of the classical bin packing. Open-ends bin packing allows pieces to be partially beyond a bin, while the classical bin packing requires all pieces to be completely inside a bin. We investigate the open-ends bin packing problem for both the off-line and on-line versions and give algorithms to solve the problem for parametric cases.
We tackle the problem of estimating the Shannon capacity of cycles of odd length. We present some strategies which allow us to find tight bounds on the Shannon capacity of cycles of various odd lengths, and suggest that the difficulty of obtaining a general result may be related to different behaviours of the capacity, depending on the “structure” of the odd integer representing the cycle length. We also describe the outcomes of some experiments, from which we derive the evidence that the Shannon capacity of odd cycles is extremely close to the value of the Lovasz theta function.
In a recent paper [1] Maynard answered a question of Harary and Manvel [2] about the reconstruction of \(square-celled \;animals\). One of his results relied on a general algebraic approach due to Alon, Caro, Krasikov, and Roditty [3]. Applying arguments of a more combinatorial nature we improve this result and give an answer to a question raised by him in [1].
Recently, in connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by \(\Phi_1, \Phi_2\), and \(\Phi_3\), have been defined [11]. It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a graph in one of the families \(\Phi_1, \Phi_2\), or \(\Phi_3\). A natural question raised from the result is whether all graphs in these families are non-Cayley. In this paper we determine the automorphism groups of all graphs in the family \(\Phi_2\). As a corollary, we show that every graph in \(\Phi_2\) is a connected non-Cayley tetravalent metacirculant graph.
Let \(G\) be a connected graph and \(S \subset E(G)\). If \(G – S\) is disconnected without isolated vertices, then \(S\) is called a restricted edge-cut of \(G\). The restricted edge-connectivity \(\lambda’ = \lambda'(G)\) of \(G\) is the minimum cardinality over all restricted edge-cuts of \(G\). A connected graph \(G\) is called \(\lambda’\)-connected, if \(\lambda'(G)\) exists. For a \(\lambda’\)-connected graph \(G\), Esfahanian and Hakimi have shown, in 1988, that \(\lambda'(G) \leq \xi(G)\), where \(\xi(G)\) is the minimum edge-degree. A \(\lambda’\)-connected graph \(G\) is called \(\lambda’\)-optimal, if \(\lambda'(G) = \xi(G)\).
Let \(G_1\) and \(G_2\) be two disjoint \(\lambda’\)-optimal graphs. In this paper we investigate the cartesian product \(G_1 \times G_2\) to be \(\lambda’\)-optimal. In addition, we discuss the same question for another operation on \(G_1\) and \(G_2\), and we generalize a recent theorem of J.-M. Xu on non \(\lambda’\)-optimal graphs.
The niche graph of a digraph \(D\) is the undirected graph defined on the same vertex set in which two vertices are adjacent if they share either a common in-neighbor or a common out-neighbor in \(D\). A hierarchy of graphs exists, depending on the condition of being the niche graph of a digraph having, respectively, no cycles, no cycles of length two, no loops, or loops. Our goal is to classify in this hierarchy all graphs of order \(n \geq 3\) having a generated subgraph isomorphic to the discrete graph on \(n – 2\) vertices.
We enumerate the bases of the bicircular matroid on \(K_{m,n}\). The structure of bases of the bicircular matroid in relation to the bases of the cycle matroid is explored. The techniques herein may enable the enumeration of the bases of bicircular matroids on larger classes of graphs; indeed one of the motivations for this work is to show the extendibility of the techniques recently used to enumerate the bases of the bicircular matroid on \(K_n\).
Motivated by the work of Granville, Moisiadis and Rees, we consider in this paper complementary \(P_3\)-packings of \(K_v\). We prove that a maximum complementary \(P_3\)-packing of \(K_v\) (with \(\lfloor\frac{v}{4} \lfloor \frac{2(v-1)}{3}\rfloor \rfloor P_3s\)) exists for all integers \(v \geq 4\), except for \(v = 9\) and possibly for \(v \in \{24, 27, 30, 33, 36, 39, 42, 57\}\).
It is proved that there is no maximal partial spread of size \(115\) in \(\mathrm{PG}(3,11)\).
In this short note, using the method developed in [10] and [11], we construct a highly symmetrical, non-simple, attractive \(7\)-Venn diagram. This diagram has the minimum number of vertices, \(21\). The only similar two, published in [1] and [11], differ from ours in many ways. One of them was found by computer search [1]. Both of them are “necklace” type Venn diagrams (see [14] for definition), but ours is not.
A graph is a unit interval graph (respectively, an \(\tilde{n}\)-graph) if we can assign to each vertex an open interval of unit length (respectively, a set of \(n\) consecutive integers) so that edges correspond to pairs of intervals (respectively, of sets) that overlap. Sakai [14] and Troxell [18] provide a linear time algorithm to find the smallest integer \(n\) so that a unit interval graph is an \(\mathbb{A}\)-graph, for the particular case of reduced connected graphs with chromatic number \(3\). This work shows how to obtain such smallest \(n\) for arbitrary graphs, by establishing a relationship with the work by Bogart and Stellpflug [1] in the theory of semiorders.
For words of length \(n\), generated by independent geometric random variables, we consider the probability that these words avoid a given consecutive \(3\)-letter pattern. As a consequence, we count permutations in \(S_n\) avoiding consecutive \(3\)-letter patterns.
A mimeomatroid is a matroid union of a matroid with itself. We develop several properties of mimeomatroids, including a generalization of Rado’s theorem, and prove a weakened version of a matroid conjecture by Rota [2].
The well-known Marriage Lemma states that a bipartite regular graph has a perfect matching. We define a bipartite graph \(G\) with bipartition \((X,Y)\) to be semi-regular if both \(x \mapsto\) deg \(x,x \in X\) and \(y \mapsto\) deg \(y, y \in Y\) are constant. The purpose of this note is to show that if \(G\) is bipartite and semi-regular, and if \(|X| < |Y|\), then there is a matching which saturates \(|X|\). (Actually, we prove this for a condition weaker than semi-regular.) As an application, we show that various subgraphs of a hypercube have saturating matchings. We also exhibit classes of bipartite graphs, some of them semi-regular, whose vertices are the vertices of various weights in the hypercube \(Q_n\), but which are not subgraphs of \(Q_n\).
The sum graph of a set \(S\) of positive integers is the graph \(G^+(S)\) having \(S\) as its vertex set, with two vertices adjacent if and only if their sum is in \(S\). A graph \(G\) is called a sum graph if it is isomorphic to the sum graph \(G^+(S)\) of some finite subset \(S\) of \(N\). An integral sum graph is defined just as the sum graph, the difference being that \(S\) is a subset of \(Z\) instead of \(N\). The sum number of a graph \(G\) is defined as the smallest number of isolated vertices when added to \(G\) results in a sum graph. The integral sum number of \(G\) is defined analogously. In this paper, we study some classes of integral sum graphs.
We say that a graph \(F\) strongly arrows \((G,H)\) and write \(F \longmapsto (G,H)\) if for every edge-coloring of \(F\) with colors red and blue, a red \(G\) or a blue \(H\) occurs as an induced subgraph of \(F\). Induced Ramsey numbers are defined by \(r^*(G,H) = \min\{|V(F)| : F \longmapsto (G,H)\}\).
The value of \(r^*(G,H)\) is finite for all graphs, and good upper bounds on induced Ramsey numbers in general, and for particular families of graphs are known. Most of these results, however, use the probabilistic method, and therefore do not yield explicit constructions. This paper provides several constructions for upper bounds on \(r^*(G,H)\), including:\(r^*(C_n) = r^*(C_n,C_n) \leq c^{(logn)^2}\), \(r^*(T,K_n) \leq |T|n^{|T|log|T|}\), \(r^*(B,C_n) \leq |B|^{\lceil log n \rceil +4}\) ,where \(T\) is a tree, \(B\) is bipartite, \(K_n\) is the complete graph on \(n\) vertices, and \(C_n\) is a cycle on \(n\) vertices. We also have some new upper bounds for small graphs: \(r^*(K_3 + e) \leq 21\), and \(r^*(K_4 – e) \leq 46\).
An \(L(2,1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x)-f(y)|\geq 2\quad\text{if}\quad d_G(x,y)=1\) and \(|f(x)-f(y)|\geq 1\quad\text{if}\quad d_G(x,y)=2\). The \(L(2,1)\)-labeling problem is to find the smallest number \(\lambda(G)\) such that there exists an \(L(2,1)\)-labeling function with no label greater than \(\lambda(G)\). Motivated by the channel assignment problem introduced by Hale, the \(L(2,1)\)-labeling problem has been extensively studied in the past decade. In this paper, we study this concept for digraphs. In particular, results on ditrees are given.
