\(\overline S_k\)-factorization of symmetric complete tripartite digraphs (Q1969801)

Let \(V_1\cup V_2\cup V_3\) be the vertex set of a symmetric complete tripartite digraph \(D\) with partite sets \(V_1,V_2,V_3\) and let \(n_i\) denote the number of vertices belonging to \(V_i\) for \(i\in\{1,2,3\}.\) An \(\overline S_k\)-factorization of \(D\) is a union of arc-disjoint spanning subgraphs \(F_t\) of \(D\) such that each component of \(F_t\) is a star on \(k\) vertices with arcs directed from a center-vertex \(c\in V_i\) to \(\frac{k-1}2\) vertices of \(V_a\) and to \(\frac{k-1}2\) vertices of \(V_b,\) where \(\{i,a,b\}=\{1,2,3\}.\) The author proves that \(D\) has an \(\overline S_k\)-factorization if and only if (i) \(k\) is odd, \(k\geq 3\) and (ii) \(n_1=n_2=n_3\) for \(k=3\) and \(n_1=n_2=n_3\equiv 0\pmod{\frac{k(k-1)}2}\) for \(k\geq 5.\)