A \((n,5)\)-cage is a minimal graph of regular degree \(n\) and girth \(5\). Let \(f(n,5)\) denote the number of vertices in a \((n,5)\)-cage. The best known example of an \((n,5)\)-cage is the Petersen graph, the \((3,5)\)-cage. The \((4,5)\)-cage is the Robertson graph, the \((7,5)\)-cage is the Hoffman-Singleton graph, the \((6,5)\)-cage was found by O’Keefe and Wong~[2] and there are three known \((5,5)\)-cages. No other \((n,5)\)-cages are known for \(n \geq 8\). In this paper, we will use a graph structure called remote edges and a set of mutually orthogonal Latin squares to give an upper bound of \(f(n,5)\) for \(n = 2^k+1\).

Let \(S\) be a set of graphs on which a measure of distance (a metric) has been defined. The distance graph \(D(S)\) of \(S\) is that graph with vertex set \(S\) such that two vertices \(G\) and \(H\) are adjacent if and only if the distance between \(G\) and \(H\) (according to this metric) is \(1\). A basic question is the determination of which graphs are distance graphs. We investigate this question in the case of a metric which we call the switching distance. The question is answered in the affirmative for a number of classes of graphs, including trees and all cycles of length at least \(4\). We establish that the union and Cartesian product of two switching distance graphs are switching distance graphs. We show that each of \(K_3\), \(K_{2,4}\) and \(K_{3,3}\) is not a switching distance graph.

A set \(\mathcal{P} \subseteq V(G)\) is a \(k\)-packing of a graph \(G\) if for every pair of vertices \(u,v \in P\), \(d(u,v) \geq k+1\). We define a graph \(G\) to be \(k\)-equipackable if every maximal \(k\)-packing of \(G\) has the same size. In this paper, we construct, for \(k \leq 1\), an infinite family \(\mathcal{F}_k\) of \(k\)-equipackable graphs, recognizable in polynomial time. We prove further that for graphs of girth at least \(4k+4\), every \(k\)-equipackable graph is a member of \(\mathcal{F}_k\).

An \(m \times n\) ideal matrix is a \(3\)-periodic \(m \times n\) binary matrix which satisfies the following two conditions: (1) each column of this matrix contains precisely one \(1\) and (2) if it is visualized as a dot pattern (with each dot representing a \(1\)), then the number of overlapping dots at all actual shifts are \(1\) or \(0\). Let \(s(n)\) denote the smallest integer \(m\) such that an \(m \times n\) ideal matrix exists. In this paper, we reduce the upper bound of \(s(n)\) which was found by Fung, Siu and Ma. Also, we list an upper bound of \(s(n)\) for \(14 \leq n \leq 100\).

I. Several unbiased tournament schedules for round robin doubles tennis are presented, in a form which can be useful to the urban league tournament director. The unbiased tournament affords less restriction than does the usual spouse-avoiding tournament (see~[{7}]). As gender considerations are not necessary, it is most often the tournament of choice.

In this note, we give a method to construct binary self-dual codes using weighing matrices. By this method, we construct extremal self-dual codes obtained from weighing matrices. In particular, the extended Golay code and new extremal singly-even codes of length \(40\) are constructed from certain weighing matrices. We also get necessary conditions for the existence of some weighing matrices.

Symmetric balanced squares for different sizes of array and for different numbers of treatments have been constructed. An algorithm, easily implementable on computers, has been developed for construction of such squares whenever the parameters satisfy the necessary conditions for existence of the square. The method of construction employs \(1\)-factorizations of a complete graph or near \(1\)-factorizations of a complete graph, depending on whether the size of the array is even or odd, respectively. For odd sized squares the method provides a solution directly based on the near \(1\)-factorization. In the case of the squares being of even size, we use Hall’s matching theorem along with a \(1\)-factorization if \([\frac{n^2}{v}]\) is even, otherwise, Hall’s matching theorem together with Fulkerson’s~\([4]\) theorem, on the existence of a feasible flow in a network with bounds on flow leaving the sources and entering the sinks, lead to the required solution.

This paper presents a probabilistic polynomial-time reduction of the discrete logarithm problem in the general linear group \(\mathrm{GL}(n, \mathbb{F})\) to the discrete logarithm problem in some small extension fields of \(\mathbb{F}_p\).