Perfect Mendelsohn Designs

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.