Star-factorization of symmetric complete bipartite multi-digraphs (Q1974543)

The symmetric complete bipartite multi-digraph \(\lambda K^{*}_{m,n}\) is the symmetric complete bipartite digraph \(K^{*}_{m,n}\) in which every arc is taken \(\lambda\) times. For \(k\geq 3\) let \(S_k\) be the orientation of the star \(K_{1,k-1}\) such that every arc in \(S_k\) is directed from the center vertex towards leaves. It is shown that \(\lambda K^{*}_{m,n}\) is the union of arc-disjoint \(S_k\) if and only if \(m=n\equiv 0\pmod {k (k-1)/d}\), where \(d=(\lambda, k-1)\).