Decomposition of the complete bipartite multigraph into cycles and stars

Let Ck denote a cycle of length k, and let Sk denote a star with k edges. For multigraphs F, G, and H, a decomposition of F is a set of edge-disjoint subgraphs of F whose union is F, and a (G,H)-decomposition of F is a decomposition of F into copies of G and H using at least one of each. In this paper, necessary and sufficient conditions for the existence of a (Ck,Sk)-decomposition of the complete bipartite multigraph are given.