On Kp,q-factorization of complete bipartite multigraphs

Let λKm,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A Kp,q-factorization of λKm,n is a set of edge-disjoint Kp,q-factors of λKm,n which partition the set of edges of λKm,n. When p = 1 and q is a prime number, Wang, in his paper [On K1,q-factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K1,q-factorization of Km,n and gave a sufficient condition for such a factorization to exist. In papers [K1,k-factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; Kp,q-factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λKm,n to have a Kp,q-factorization. As a special case, it is shown that the necessary condition for the Kp,q-factorization of λKm,n is always sufficient when p : q = k : (k + 1) for any positive integer k.