Ǩp,q-factorization of symmetric complete tripartite digraphs

Let Kn1,n2,n3∗ denote the symmetric complete tripartite digraph with partite sets V1,V2,V3 of n1,n2,n3 vertices each, and let Ǩp,q denote the complete bipartite digraph in which all arcs are directed away from p start-vertices in Vi to q end-vertices in Vj with {i,j}⊂{1,2,3}. We show that a necessary condition for the existence of a Ǩp,q-factorization of Kn1,n2,n3∗ is n1=n2=n3≡0 (mod dp′q′(p′+q′)) for p′+q′≡1,2 (mod 3) and n1=n2=n3≡0 (mod dp′q′(p′+q′)/3), 2n1⩾pp′, 2n1⩾qq′ for p′+q′≡0 (mod 3), where d=(p,q), p′=p/d, q′=q/d. Several sufficient conditions are also given.