Resolvable perfect Mendelsohn designs with block size five
A \((v,k,\lambda)\)-Mendelsohn design (MD) is, using graph-theoretic notation, a decomposition of the complete directed multigraph \(\lambda\text{DK}_v\) on \(v\) vertices into \(k\)-circuits. A perfect Mendelsohn design has the additional property that for any \(r\), \(1 \leq r \leq k-1\), and for any two distinct vertices \(x\) and \(y\), there are exactly \(\lambda\) of the \(k\)-circuits along which the (directed) distance from \(x\) to \(y\) is \(r\). A Mendelsohn design is said to be resolvable if the \(k\)-circuits can be partitioned into sets of \(k\)-circuits that in turn partition the vertex set. A resolvable perfect \((v,k,\lambda)\)-MD is called a \((v,k,\lambda)\)-RPMD. It is shown that the necessary condition for the existence of a \((v,5,1)\)-RPMD, namely that \(v\) is divisible by 5, is sufficient except for \(v \in \{5,10\}\) and with the 17 cases \(v \in \{20,30,90,95,100,110,115,120,130,135,140,150,160,170,190,195,210\}\) unresolved.
scientific article; zbMATH DE number 1746224