Star-factorization of the Complete Bipartite Multigraphs

Let λKm,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A Kp,q-factorization of λKm,n is a set of Kp,q-factors of λKm,n which partition the set of edges of λKm,n. When λ = 1, Martin, in [Complete bipartite factorizations by complete bipartite graphs, Discrete Math., 167/168 (1997), 461–480], gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will study the Kp,q-factorization of λKm,n for p = 1, to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.