Some gregarious kite decompositions of complete equipartite graphs

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 ≡ 0, 1 (mod k), or k is odd and m ≡ 0 (mod k). As a consequence, we prove necessary and sufficient conditions for decomposing complete equipartite graphs into gregarious cycles of prime length.