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

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. For c >= 2 we prove that a regular c-partite tournament with r >= 2 vertices in each partite set contains a directed path with exactly two vertices from each partite set. Furthermore, if c >= 4, then we will show that almost all regular c-partite tournaments D contain a directed path with exactly r - s vertices from each partite set for each given integer s is an element of N, if r is the cardinality of each partite set of D. Some related results are also presented. (c) 2006 Elsevier B.V. All rights reserved.