A tournament design, \({TD}(n, c)\), is a \(c\)-row array of the \(\binom{n}{2}\) pairs of elements from an \(n\)-set such that every element appears at most once in each column and there are no empty cells. An interval balanced tournament design, \(\text{IBTD}(n, c)\), satisfies the added condition that the appearances of each element are equitably distributed amongst the columns of the design. We settle the existence question for all \(\text{IBTD}(n,c)\)s by showing that they can be constructed for all admissible parameters and discuss the application of \(\text{IBTD}\)s to scheduling round robin tournaments fairly with respect to the amount of rest allocated to each participant.

We provide two new upper bounds on the total chromatic number of all hypergraphs and give two conjectures related to both the Total Colouring Conjecture for graphs and the Erdős-Faber-Lovász Conjecture.

The first serious mathematical study of whist tournament designs was carried out in the 1890s by E.H. Moore. In this survey, I shall outline briefly the subsequent work which culminated in the proof of the existence of whist tournaments of all possible orders by Baker, Wilson, and Hanani in the 1970s, and then describe some more recent work, mainly by N.J. Finizio, Y.S. Liaw, and the author, on the construction of cyclic whist tournaments. In particular, triple whist tournaments will be discussed.

A graph \(G\) on \(n\) vertices is \({pancyclic}\) if \(G\) contains cycles of all lengths \(\ell\) for \(3 \leq \ell \leq n\) and \(G\) is \({cycle \; extendable}\) if for every non-hamiltonian cycle \(C \subset G\) there is a cycle \(C’ \subset G\) such that \(V(C) \subset V(C’)\) and \(|V(C’) \setminus V(C)| = 1\). We prove that

1. every \(2\)-connected \(K_{1,3}\)-free graph is pancyclic, if \(G\) is \(P_5\)-free and \(n \geq 6\), if \(G\) is \(P_6\)-free and \(n \geq 10\), or if \(G\) is \(P_7\)-free, deer-free and \(n \geq 14\), and
2. every \(2\)-connected \(K_{1,3}\)-free and \(Z_2\)-free graph on \(n \geq 10\) vertices is cycle extendible using at most two chords of the cycle.

The problem of finding the distance between two graphs is known to be NP-complete. In this paper, we describe a heuristic algorithm that uses simulated annealing to find an upper bound for the distance between two graphs. One of the motivations for developing such an algorithm comes from our interest in finding the diameter of families of non-isomorphic extremal graphs. We tested our algorithm on each family of extremal graphs with up to \(16\) vertices and show that the exact distance was obtained in all cases.

\(Z\)-cyclic whist tournaments for \(q+1\) players, \({Wh}(q+1)\), where \(q\) is a prime, \(q \equiv 3 \pmod{4}\), are quite rare. Solutions for \(q = 3, 7, 11, 19, 23,\) and \(31\) were known in the early to mid 1890’s. Since that time no new such \({Wh}(q +1)\) have appeared.
Here we present \(Z\)-cyclic \({Wh}(q + 1)\) for \(q = 43, 47, 59\). Also presented for the first time is a \(Z\)-cyclic \({Wh}(45)\) and a \(Z\)-cyclic \({Wh}(40)\) that has the three person property. All of these results were obtained via the computer.

There are many graphs with the property that every subgraph of a given simple isomorphism type can be completed to a larger subgraph which is embedded in its ambient parent graph in a nice way. Often, such graphs can be classified up to isomorphism. Here we survey theorems on polar space graphs, graphs with the cotriangle property, copolar graphs, Fischer spaces, and generalized Fischer spaces, as well as graphs with the odd coclique property.

Permutation graphs, a well-known class of perfect graphs, has attracted the attention of numerous researchers. There are two noteworthy representations of permutation graphs. Permutation diagrams have been widely employed in theoretical and application research. The \(2\)-dimensional Euclidean representation suggested by Ore is relatively unknown and unexplored. In this paper, we demonstrate the utility of the latter representation in the investigation of the Hamiltonian Path problem in permutation graphs.

In this paper, we investigate the relationship between the profiles of Hadamard matrices and the weights of the doubly even self-orthogonal/dual \([n, m, d]\) codes from Hadamard matrices of order \(n = 8t\) with \(t \geq 1\). We show that such codes have \(m \leq \frac{n}{2}\), and give some computational results of doubly even self-orthogonal/dual \([n,m,d]\) codes from Hadamard matrices of order \(n = 8t\), with \(1 \leq t \leq 9\).

Let \(G\) be a finite strongly connected mixed graph (i.e., a graph with both undirected and directed edges, in which each vertex can be reached from every other vertex if directed edges can only be traversed in their direction of orientation). We establish a necessary and sufficient condition for it to be possible to transform some undirected edges of \(G\) into directed edges so that each vertex becomes the head of a prescribed number of newly directed edges and \(G\) remains strongly connected. A special case of this result yields a new proof (not requiring matroid techniques) of a necessary and sufficient condition for it to be possible to split each vertex of a finite connected graph into a prescribed number of vertices whilst preserving connectedness.