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 \(M > N \geq 2\), 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 \(\lambda K_v^*\) on \(v\) vertices into \(k\)-circuits such that for any \(r\), \(1 \leq r \leq k-1\), 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 \(n \geq 7\) 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 \(bm – na \geq 2(n-b)\) and \(0 \leq a < b \leq n\).

The chromatic polynomial captures a good deal of combinatorial information about a graph, describing its acyclic orientations, its all-terminal reliability, its spanning trees, as well as its colorings. Several methods for computing the chromatic polynomial of a graph G construct a computation tree for G whose leaves are “simple” base graphs for which the chromatic polynomial is readily found. Previously studied methods involved base graphs which are complete graphs, completely disconnected graphs, forests, and trees. In this paper, we consider chordal graphs as base graphs. Algorithms for computing the chromatic polynomial based on these concepts are developed, and computational results are presented.