Complete bipartite factorisations by complete bipartite graphs

We study complete Kp,q-factorisations of Km, n. Simple necessary conditions are found and we conjecture that these conditions are also sufficient. A general construction is given to find infinite families of factorisations proving the conjecture in many cases. The conjecture is proved for Kl,q-factorisations of Kn, n except for q = 4k + 1. The conjecture is also proved for a further family of K1, 3-factorisations.