Kp,q-factorization of complete bipartite graphs

Abstract

Let K m , n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K p , q -factorization of K m , n is a set of edge-disjoint K p , q -factors of K m , n which partition the set of edges of K m , n . When p = 1 and q is a prime number, Wang, in his paper “On K 1, k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K 1, q -factorization of K m , n and gave a sufficient condition for such a factorization to exist. In the paper “ K 1, k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang's result to the case that q is any positive integer. In this paper, we give a sufficient condition for K m , n to have a K p , q -factorization. As a special case, it is shown that the Martin's BAC conjecture is true when p : q = k :( k +1) for any positive integer k .