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 \(0 \leq a < b \leq n\).