Ĉk-factorization of symmetric complete bipartite and tripartite multi-digraphs

We show that a necessary and sufficient condition for the existence of a Ĉk-factorization of the symmetric complete bipartite multi-digraph λKn1, n2∗ is (i) k≡0 (mod 2) and (ii) n1=n2≡0 (mod k/2). We also show that a necessary and sufficient condition for the existence of a Ĉk-factorization of the symmetric complete tripartite multi-digraph λKn1,n2, n3∗ is (i) k≡0 (mod 2) and (ii) n1=n2=n3≡0 (mod k).