On resolvable mixed path designs (Q923106)

An \(H\)-factor of a multigraph \(G\) is a spanning subgraph each component of which is isomorphic to \(H\). Let \(P_ k\) denote the path with \(k\) vertices and \(\lambda K_ n\) denote the complete multigraph with \(n\) vertices in which every pair of distinct vertices is joined by \(\lambda\) edges. The authors prove that the edge set of \(\lambda K_ n\) has a partition into \(s\) 1-factors and \(t\) \(P_ k\)-factors, \(k\geq 2\) and \(st\neq 0\), if and only if \(n\equiv 0 \pmod 2\), \(n\equiv 0\pmod k\) and \(ks+2t(k-1)=\lambda k(n-1)\).