Resolvable gregarious cycle decompositions of complete equipartite graphs (Q924973)

Let \(K_{n(m)}\) be a complete equipartite graph with \(n\) parts of size \(m\) each, and \(C_n\) be a cycle with \(n\) vertices. When \(C_n\) is a subgraph of \(K_{n(m)}\), it is called gregarious if it has one vertex in each partite set of \(K_{n(m)}\). A gregarious \(C_n\)-decomposition of \(K_{n(m)}\) is a collection \(\mathcal G\) of gregarious cycles \(C_n\) in \(K_{n(m)}\), such that every edge of \(K_{n(m)}\) belongs to exactly one \(C_n\in \mathcal G\). Such a decomposition is resolvable if \(\mathcal G\) is a collection of graphs isomorphic to \(mC_n\) (\(m\) vertex-disjoint copies of \(C_n\)). It is shown that a resolvable gregarious \(C_n\)-decomposition of \(K_{n(m)}\) exists for all \(m\) and \(n\) except when \(m\) is odd and \(n\) is even, or when \(n=3\) and \(m=2\) or 6.