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.