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.