On Kp,q -FACTORIZATION OF COMPLETE BIPARTITE MULTIGRAPHS

Let lambda K-m,K-n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K-p,K-q-factorization of lambda K-m,K-n is a set of edge-disjoint K-p,K-q-factors of lambda K-m,K-n which partition the set of edges of lambda K-m,K-n. When p = 1 and q is a prime number, Wang, in his paper [On K-1,K-q-factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364, investigated the K-1,K-q-factorization of K-m,K-n and gave a sufficient condition for such a factorization to exist. In papers [K-1,K-k-factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),, K-p,K-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, lambda K-m,K-n to have a K-p,K-q-factorization. As a special case, it is shown that the necessary condition for the K-p,K-q-factorization of, lambda K-m,K-n is always sufficient when p : q = k : (k + 1) for any positive integer k.