${\overrightarrow{P}}_{2k+l}$-Factorization of Symmetric Complete Bipartite Multi-Digraphs

Abstract

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