K1, pq-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)] investigated the K1,p-factorization of Km,n and gave a sufficient condition for such a factorization to exist. Du [Discrete Math. 187 (1998) and Appl. Math. J. Chinese Univ. 17B (2001)] extended Wang’s result to the case k is a prime power p. In this paper, it is shown that the conclusion in Wang’s 1994 paper is true for any prime product pq. We will give a sufficient condition for the existence of the K1,pq-factorization of Km,n, whenever p and q are prime numbers, that is (1) m ≤ pqn, (2) n ≤ pqm, (3) pqm − n ≡ pqn − m ≡ 0 (mod (pq − 1)) and (4) (pqm − n)(pqn−m) ≡ 0 (mod pq(pq − 1)(pq − 1)(m+ n)).