Cycle-factorization of symmetric complete multipartite digraphs

First, we show that a necessary and sufficient condition for the existence of a C3-factorization of the symmetric tripartite digraph Kn1,n2,n3∗, is n1 = n2 = n3. Next, we show that a necessary and sufficient condition for the existence of a C̄2k-factorization of the symmetric complete multipartite digraph Kn1, n2,…,nm is n1 = n2 = … = nm = 0 (mod k) for even m and n1 = n2 = … = ≡ 0 (mod 2k) for odd m.