Cycles with a given number of vertices from each partite set in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d(+)(x) and d(-)(x) the outdegree and the indegree of x, respectively. A digraph D is called regular, if there is a number p is an element of N such that d(+) (x) = d(-)(x) = p for all vertices x of D. A c-partite tournament is an orientation of a complete c-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether c-partite tournaments with r vertices in each partite set contain a cycle with exactly r - 1 vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if c = 2. If c >= 3, then we will show that a regular c-partite tournament with r >= 2 vertices in each partite set contains a cycle with exactly r - 1 vertices from each partite set, with the exception of the case that c = 4 and r = 2.