Decomposition of the complete bipartite graph with a 1-factor removed into cycles and stars

Let CkCk denote a cycle of length kk, and let SkSk denote a star with kk edges. For graphs FF, GG, and HH, a (G,H)-decomposition(G,H)-decomposition of FF is a decomposition of FF into copies of GG and HH using at least one of each. In this paper, necessary and sufficient conditions for the existence of a (Ck,Sk)(Ck,Sk)-decomposition of the complete bipartite graph with a 1-factor removed are given.