Outpaths of arcs in multipartite tournaments

Ak-outpath of an arcxy in a multipartite tournament is a directed path with lengthk starting fromxy such thatx does not dominate the end vertex of the directed path. This concept is a generalization of a directed cycle. We show that ifT is an almost regularn-partite (n≥8) tournament with each partite set having at least two vertices, then every arc ofT has ak-outpath for allk, 3≤k≤n−1. This project is supported by the National Natural Science Foundation of China and NSFJC.