Let $\mathcal{G}(n,m)$ denote the class of simple graphs on $n$ vertices and $m$ edges, and let $G\in \mathcal{G}(n,m)$. For suitably restricted values of $m$, $G$ will necessarily contain certain prescribed subgraphs such as cycles of given lengths and complete graphs. For example, if $m>\frac{1}{4}{n}^2$, then $G$ contains cycles of all lengths up to $\lfloor \frac{1}{2}(n+3)\rfloor$. Recently, we have established a number of results concerning the existence of certain subgraphs (cliques and cycles) in the subgraph of $G$ induced by the vertices of $G$ having some prescribed minimum degree. In this paper, we present some further results of this type. In particular, we prove that every $G\in \mathcal{G}(n,m)$ contains a pair of adjacent vertices each having degree (in $G$) at least $f(n,m)$ and determine the best possible value of $f(n,m)$. For $m>\frac{1}{4}{n}^2$, we find that $G$ contains a triangle with a pair of vertices satisfying this same degree restriction. Some open problems are discussed.

A weighing matrix $A=A(n,k)$ of order $n$ and weight $k$ is a square matrix of order $n$, with entries $0,\pm 1$ which satisfies $A{A}^T=k{I}_n$. H.C. Chan, C.A. Rodger, and J. Seberry “On inequivalent weighing matrices, $ArsCombinatoria$, $(1986)21-A,299-333$” showed that there were exactly $5$ inequivalent weighing matrices of order $12$ and weight $4$ and exactly $2$ inequivalent matrices of weight $5$. They showed that the weighing matrices of order $12$ and weights $2,3$, and $11$ were unique. Q.M. Husain “On the totality of the solutions for the symmetric block designs: $\lambda =2,k=5$ or $6$,” Sanky$\overline{a}$ $7(1945),204-208$” had shown that the Hadamard matrix of order $12$ (the weighing matrix of weight $12$) is unique. In this paper, we complete the classification of weighing matrices of order $12$ by showing that there are seven inequivalent matrices of weight $6$, three of weight $7$, six of weight $8$, four of weight $9$, and four of weight $10$. These results have considerable implications for inequivalence results for orders greater than 12.

Informally, a $(t,w,v;m)$-threshold scheme is a way of distributing partial information (chosen from a set of $v$ shadows) to $w$ participants, so that any $t$ of them can easily calculate one of $m$ possible keys, but no subset of fewer than $t$ participants can determine the key. A perfect threshold scheme is one in which no subset of fewer than $t$ participants can determine any partial information regarding the key. In this paper, we study the number $M(t,w,v)$, which denotes the maximum value of $m$ such that a perfect $(t,w,v;m)$-threshold scheme exists. It has been shown previously that$M(t,w,v)\le (v-t+1)/(w-t+1)$, with equality occurring if and only if there is a Steiner system $S(t,w,v)$ that can be partitioned into Steiner systems $S(t-1,w,v)$. In this paper, we study the numbers $M(t,w,v)$ in some cases where this upper bound cannot be attained. Specifically, we determine improved bounds on the values $M(3,3,v)$ and $M(4,4,v)$.

A triple system $B[3,\lambda ;v]$ is indecomposable if it is not the union of two triple systems $B[3,{\lambda }_1;v]$ and $B[3,{\lambda }_2;v]$ with $\lambda ={\lambda }_1+{\lambda }_2$. We prove that indecomposable triple systems with $\lambda =6$ exist for $v=8,14$ and for all $v\ge 17$.

Given a convex lattice polygon with $g$ interior lattice points, we find upper and lower bounds for the perimeter, diameter, and width of the polygon. For small $g$, the extremal figures were found by computer.

We survey the existence of base sequences, that is, four sequences of lengths $m+p,m+p,m,m,p$ odd with zero auto-correlation function, which can be used with Yang numbers and four disjoint complementary sequences (and matrices) with zero non-periodic (periodic) auto-correlation function to form longer sequences. We survey their application to make orthogonal designs $OD(4t;t,t,t,t)$. We give the method of construction of $OD(4t;t,t,t,t)$ for $t=1,3,\dots ,41,45,\dots ,65,67$, $69,75,77,81,85,87,91,93,95,99,101,105$, $111,115,117,119,123,125,129,133,141,\dots ,147,153$, $155,159,161,165,169,171,175,177,183,185,189,195,201,203,205,209$.

In this note we construct designs on the hexagonal grid to be used for choice experiments.

Digraph $D$ is defined to be exclusive $(M,N)$-transitive if, for each pair of vertices $x$ and $y$, for each $xy$-path $P_1$ of length $M$, there is an $xy$-path $P_2$ of length $N$ such that $P_1\cap P_2=\{x,y\}$. It is proved that computation of a minimal edge augmentation to make $K$ exclusive $(M,N)$-transitive is NP-hard for $M>N\ge 2$, even if $D$ is acyclic. The corresponding decision problems are NP-complete. For $N=1$ and $D=(V,E)$ with $|V|=n$, an $O(n^{M+3})$ algorithm to compute the exclusive $(M,1)$-transitive closure of an arbitrary digraph is provided.

Let $v$, $k$, and $\lambda$ be positive integers. A perfect Mendelsohn design with parameters $v$, $k$, and $\lambda$, denoted by $(v,k,\lambda )$-PMD, is a decomposition of the complete directed multigraph $\lambda K_v^{\ast }$ on $v$ vertices into $k$-circuits such that for any $r$, $1\le r\le k-1$, and for any two distinct vertices $x$ and $y$ there are exactly $\lambda$ circuits along which the (directed) distance from $x$ to $y$ is $r$. In this survey paper, we describe various known constructions, new results, and some further questions on PMDs.

A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. It is proved in this paper that there are three pairwise orthogonal diagonal Latin squares of order $n$ for all $n\ge 7$ with $28$ possible exceptions, in which $118$ is the greatest one.

A graph $G$ is $[a,b]$-covered if each edge of $G$ belongs to an $[a,b]$-factor. Here, a necessary and sufficient condition for a graph to be $[a,b]$-covered is given, and it is shown that an $[m,n]$-graph is $[a,b]$-covered if $bm–na\ge 2(n-b)$ and $0\le a<b\le n$.