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 [email protected]?N, if r is the cardinality of each partite set of D. Some related results are also presented.

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