Applying Glauberman’s \(Z^*\)-theorem, it is shown that every finite group \(G\) is strongly \(P_3\)-sequenceable, i.e. there exists a sequencing \((x_1,\ldots,x_{N})\) of the elements of \(G\setminus\{1\}\), such that all products \(x_ix_{i+1}x_{i+2}\) (\(1\leq i\leq N-2\)), \(x_{N-1}x_{N}x_{1}\) and \(x_Nx_{1}x_2\) are nontrivially rewritable. This was conjectured by J. Nielsen in~[N].

Competition graphs were first introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices (except \(K_2\)) which is the competition graph of a loopless digraph is also the competition graph of a strongly connected digraph. We establish an analogous result for one generalization of competition graphs, the \(p\)-competition graph. Furthermore, we establish some large classes of graphs, including trees, as the \(p\)-competition graph of a loopless Hamiltonian digraph and show that interval graphs on \(n \geq 4\) vertices are the \(2\)-competition graphs of loopless Hamiltonian digraphs.

Let \(H_n < S_n\), where \(H_n\) is a Sylow \(p\)-subgroup of \(S_n\), the symmetric group on \(n\) letters. Let \(A_n\) denote the number of derangements in \(H_n\), and \(f_n = \frac{h_n}{|H_n|}\). We will show that the sequence \(\{f_n\}_{n=1}^{\infty}\) is dense in the unit interval, but is Cesàro convergent to \(0\).

Let \(B(G)\) and \(B_c(G)\) denote the bandwidth and cyclic bandwidth of graph \(G\), respectively. In this paper, we shall give a sufficient condition for graphs to have equal bandwidth and cyclic bandwidth. This condition is satisfied by trees. Thus all theorems on bandwidth of graphs apply to cyclic bandwidth of graphs satisfying the sufficiency condition, and in particular, to trees. We shall also give a lower bound of \(B_c(G)\) in terms of \(B(G)\).

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 \(14 \leq n \leq 100\).