S̄k-factorization of symmetric complete tripartite digraphs

We show that a necessary and sufficient condition for the existence of an S̄k-factorization of the symmetric complete tripartite digraph Kn1, n2, n3∗ is (i) k is odd, k⩾3 and (ii) n1=n2=n3 for k=3 and n1=n2=n3≡0 (mod k(k−1)/2) for k⩾5.