Šk-factorization of symmetric complete tripartite digraphs

We show that a necessary condition for the existence of an Šk-factorization of the symmetric complete tripartite digraph Kl,m,n∗ is l = m = n ≡ 0 (mod k(k − 1)) for k = 1, 2 (mod 3) and l = m = n ≡ 0 (mod k(k − 1)/3), 2n ⩾ (k − 1)2 for k ≡ 0 (mod 3). Several sufficient conditions are also given.