${\hat{S}}_k$-Factorization of Complete Multipartite Symmetric Digraphs

Abstract

In this paper, we focus our study on finding necessary and sufficient conditions required for the existence of an ${\hat{S}}_k$-factorization of $(K_m\circ {\overline{K}}_n{)}^{\ast }$ and $(C_m\circ {\overline{K}}_n{)}^{\ast }$. In particular, we show that the necessary conditions for the existence of an ${\hat{S}}_k$-factorization of $(K_m\circ {\overline{K}}_n{)}^{\ast }$ are sufficient except when none of $m$ or $n$ is a multiple of $k$. In fact, our results deduce some of the results of Ushio on ${\hat{S}}_k$-factorizations of complete bipartite and tripartite symmetric digraphs.