K1, k-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 1, k -factorization of K m, n is a set of edge-disjoint K 1, k -factors of K m, n which partition the set of edges of K m, n . When k is a prime number p, Wang (Discrete Math. 126 (1994) 359) investigated the K 1, p -factorization of K m, n and gave a sufficient condition for such a factorization to exist. Later, there are many work to extend Wang's result to the case k is a positive integer. In papers (Discrete Math. 187 (1998) 273; Appl. Math. JCU 17B (2001) 107), Du extended Wang's result to the case k is a prime power p u . In paper (Austral. J. Combin. 26 (2002) 85) for a prime product pq, Du investigated K 1, pq -factorization of K m, n and gave a sufficient condition for such a factorization to exist. In this paper, it is shown that the conclusion in (Discrete Math. 126 (1994) 359) is true for any positive integer k. We shall prove that a sufficient condition for the existence of the K 1, k -factorization of K m, n , whenever k is any positive integer, is that (1) m⩽ kn, (2) n⩽ km, (3) km−n≡kn−m≡0 ( mod (k 2−1)) and (4) (km−n)(kn−m)≡0 ( mod k(k−1)(k 2−1)(m+n)) .