K1,k-factorization of complete bipartite graphs

Let Km,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A K1,k-factorization of Km,n is a set of edge-disjoint K1,k-factors of Km,n which partition the set of edges of Km,n. When k is a prime number p, Wang (Discrete Math. 126 (1994) 359) investigated the K1,p-factorization of Km,n and gave a sufficient condition for such a factorization to exist. Later, there are many work to extend Wangʹs result to the case k is a positive integer. In papers (Discrete Math. 187 (1998) 273; Appl. Math. JCU 17B (2001) 107), Du extended Wangʹs result to the case k is a prime power pu. In paper (Austral. J. Combin. 26 (2002) 85) for a prime product pq, Du investigated K1,pq-factorization of Km,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 K1,k-factorization of Km,n, whenever k is any positive integer, is that (1) m⩽kn, (2) n⩽km, (3) km−n≡kn−m≡0 (mod (k2−1)) and (4) (km−n)(kn−m)≡0 (mod k(k−1)(k2−1)(m+n)).