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.