\(\widehat K_{p,q}\)-factorization of symmetric complete tripartite digraphs (Q5937610)

Let \(K^*_{n_1,n_2,n_3}\) be the symmetric complete tripartite digraph with partite sets of cardinalities \(n_1,n_2\), and \(n_3\). Furthermore, let \(\widehat K_{p,q}\) be the complete bipartite digraph with the partite sets \(V_1,V_2\), such that \(|V_1|=p\), \(|V_2|=q\) with the property that all arcs are oriented from \(V_1\) to \(V_2\). The authors present a simple necessary and a couple of sufficient conditions for the existence of a \(\widehat K_{p,q}\)-factorization of the special digraph \(K^*_{n_1,n_2,n_3}\). The procedures are routine.