Star-factorization of the Complete Bipartite Multigraphs

Let λKm,nbe a complete bipartite multigraph with two partite sets having m and n vertices,respectively. A Kp,q-factorization of λKm,nis a set of Kp,q-factors of λKm,nwhich 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,nfor p = 1,to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.