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 ≥ 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 P_k+₁ and q copies of S_k+ ₁ if and only if k(p + q) = (ⁿ₂).