Outpaths of arcs in regular 3-partite tournaments

Guo [Outpaths in semicomplete multipartite digraphs, Discrete Appl. Math. 95 (1999) 273–277] proposed the concept of the outpath in digraphs. An outpath of a vertex x (an arc xy, respectively) in a digraph is a directed path starting at x (an arc xy, respectively) such that x does not dominate the end vertex of this directed path. A k-outpath is an outpath of length k. The outpath is a generalization of the directed cycle. A c-partite tournament is an orientation of a complete c-partite graph. In this paper, we investigate outpaths of arcs in regular 3-partite tournaments. We prove that every arc of an r-regular 3-partite tournament has 2(when r ≥ 1), 3(when r ≥ 2), and 5-, 6-outpaths (when r ≥ 3). We also give the structure of an r-regular 3-partite tournament D with r ≥ 2 that contains arcs which have no 4-outpaths. Based on these results, we conjecture that for all k ∈ {1, 2, . . . , r − 1}, every arc of r-regular 3-partite tournaments with r ≥ 2 has (3k−1)and 3k-outpaths, and it has a (3k+1)outpath except an r-regular 3-partite tournament.