Decomposition of complete bipartite graphs into paths and stars with same number of edges

In this paper, we give some necessary and sufficient conditions for decomposing the complete bipartite graph Km,n into paths Pk+1 and stars Sk+1 with k edges each. In particular, we find necessary and sufficient conditions for accomplishing this when m>k and n≥3k, and we give a complete solution when k=3. We also apply the results to show that for nonnegative integers p and q and positive integers n and k with n≥4k, there exists a decomposition of the complete graph Kn into p copies of Pk+1 and q copies of Sk+1 if and only if k(p+q)=n2.