Abstract
Let K m,nbe 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,nand 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.
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Du, B., Wang, J. K p,q-factorization of complete bipartite graphs. Sci. China Ser. A-Math. 47, 473–479 (2004). https://doi.org/10.1360/02ys0360
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DOI: https://doi.org/10.1360/02ys0360