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

In this paper, we focus our study on finding necessary and sufficient conditions required for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) and \( (C_m \circ \overline{K}_n)^* \). In particular, we show that the necessary conditions for the existence of an \( \hat{S}_k \)-factorization of \( (K_m \circ \overline{K}_n)^* \) 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.