On K 1,k -factorization of bipartite multigraphs

A K _1,k -factorization of λK _m,n is a set of edge-disjoint K _1,k -factors of λK _m,n , which partition the set of edges of λK _m,n . In this paper, it is proved that a sufficient condition for the existence of K _1,k -factorization of λK _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) λ(km − n)(kn − m) ≡ 0 (mod k(k − 1)(k ² − 1)(m + n)). Supported by the National Natural Science Foundation of China (10571133)