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

Let C k denote a cycle of length k , and let S k denote a star with k edges. For graphs F , G , and H , a ( G , H ) - d e c o m p o s i t i o n 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 ( C k , S k ) -decomposition of the complete bipartite graph with a 1-factor removed are given.