Common multiples of complete graphs and a 4-cycle

A graph G is a common multiple of two graphs H1 and H2 if there exists a decomposition of G into edge-disjoint copies of H1 and also a decomposition of G into edge-disjoint copies of H2. In this paper, we consider the case where H1 is the 4-cycle C4 and H2 is the complete graph with n vertices Kn. We determine, for all positive integers n, the set of integers q for which there exists a common multiple of C4 and Kn having precisely q edges.