\(\vec{P}_{2k+l}\)-Factorization of Symmetric Complete Bipartite Multi-Digraphs

Abstract

Let \(\vec{P_l}\) be the directed path on \(r\) vertices and \(\lambda K^*_{m,n}\) be the symmetric complete bipartite multi-digraph with two partite sets having \(m\) and \(n\) vertices. A \(\vec{P_l}\)-factorization of \(\lambda K^*_{m,n}\) is a set of arc-disjoint \(\vec{P_l}\)-factors of \(\lambda K^*_{m,n}\), which is a partition of the set of arcs of \(\lambda K^*_{m,n}\). In this paper, it is shown that a necessary and sufficient condition for the existence of a \(\vec{P}_{2k+l}\)-factorization of \(\lambda K^*_{m,n}\) for any positive integer \(k\).