Some Gregarious Cycle Decompositions of Complete Equipartite Graphs

Abstract

A $k$-cycle decomposition of a multipartite graph $G$ is said to be gregarious if each $k$-cycle in the decomposition intersects $k$ distinct partite sets of $G$. In this paper we prove necessary and sufficient conditions for the existence of such a decomposition in the case where $G$ is the complete equipartite graph, having $n$ parts of size $m$, and either $n\equiv 0,1\pmod{k}$, or $k$ is odd and $m\equiv 0\pmod{k}$. As a consequence, we prove necessary and sufficient conditions for decomposing complete equipartite graphs into gregarious cycles of prime length.