Evenly partite star-factorization of symmetric complete tripartite multi-digraphs

We show that a necessary and sufficient condition for the existence of an S2q+1 - factorization of the symmetric complete tripartite multi-digraph λKn1, n2, n3∗ is (i) n1 = n2 = n3 for q = 1 and (ii) n1 = n2 = n3 = 0 (mod (2q+1)q/d) for q ≥ 2, where d = (λ, q).