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 K m , n into paths P k + 1 and stars S k + 1 with k edges each. In particular, we find necessary and sufficient conditions for accomplishing this when m > k and n ≥ 3 k , 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 ≥ 4 k , there exists a decomposition of the complete graph K n into p copies of P k + 1 and q copies of S k + 1 if and only if k ( p + q ) = n 2 .