\(\widehat S_k\)-factorization of symmetric complete tripartite digraphs
Let \(K^*_{l,m,n}\) be the symmetric complete tripartite digraph with partite sets of \(l,m\) and \(n\) vertices. If \(K^*_{l,m,n}\) is expressed as an arc-disjoint sum of \(\widehat S_k\)-factors, then this sum is called an \(\widehat S_k\)-factorization of \(K^*_{l,m,n}.\) The author shows that a necessary condition for the existence of an \(\widehat S_k\)-factorization is \(l=m=n\equiv 0\pmod {k(k-1)}\), when \(k\equiv 1,2\pmod 3\) and \(l=m=n\equiv 0\pmod {k(k-1)/3}\), \(2n\geq(k-1)^2,\) when \(k\equiv 0 \pmod 3\). The author also gives several sufficient conditions.
scientific article; zbMATH DE number 1322055