All graphs meeting the basic necessary conditions to be the leave graph of a maximal partial triple system with at most thirteen elements are generated. A hill-climbing algorithm is developed to determine which of these candidates are in fact leave graphs. Improved necessary conditions for a graph to be a leave graph are developed.

Some new lower bounds for higher Ramsey numbers are presented. Results concerning generalized hypergraph Ramsey numbers are also given.

We enumerate the perfect one-factorizations of \(K_{50}\), which are generated by starters in \({Z}_{49}\), fixed by multiplication by \(18\) and \(30\). There are precisely \(67\) non-isomorphic examples.

Let the vertices of a graph denote computer processes which communicate by passing messages along edges. It has been a standard Computer Science problem to provide algorithms that let the processes solve problems jointly (e.g. leader election, clock synchronization). What if some of the processes are maliciously faulty, i.e. send messages calculated to sabotage joint algorithms? Here we review a few “byzantine agreement” algorithms with interesting graph-theoretic features and raise questions about graph connectivity and diameter (with a few answers).

Let the vertices of a graph denote processes in a distributed or time-shared computer system; let two vertices be connected by an edge if the two processes cannot proceed at the same time (they mutually exclude one another). Managing mutual exclusion and related scheduling problems has given rise to substantial literature in computer science. Some methods of attack include covering or partitioning the graph with cliques or threshold graphs. Here I survey some recent graph-theoretic results and examples motivated by this approach.

A triangle in a Steiner triple system \(S\) is a triple of blocks from \(S\) which meet pairwise and whose intersection is empty. If \(S\) contains \(b\) blocks, and \(b = 3q + 8\), where \(0 \leq 8 \leq 2\), then a triangulation of \(S\) is a collection of \(q\) triangles \(\{T_1, T_2, \ldots, T_q\}\) in \(S\) such that no two distinct triangles share a common block. It is shown that, for \(v \equiv 1\) or \(3 \pmod{6}\), there exists a Steiner triple system which admits a triangulation. Moreover, if \(8 = 2\), there is a triangulated triple system in which the pair of blocks not occurring in a triangle are disjoint, and a triangulated triple system in which they intersect.