Outpaths in semicomplete multipartite digraphs

Abstract

An outpath of a vertex x (an arc xy, respectively) in a digraph is a directed path starting at x ( xy, respectively) such that x dominates the endvertex of the path only if the endvertex also dominates x. Firstly, we show that if D is a strongly connected semicomplete n-partite ( n⩾3) digraph, then every vertex v of D has an outpath of length k−1 for all k∈{3,4,…, n}. Our result generalizes a theorem of Moon (Canad. Math. Bull. 9 (1966) 297–301) for tournaments. Secondly, we show that if T is a regular n-partite ( n⩾3) tournament, then every arc of T has an outpath of length k−1 for all k satisfying 3⩽ k⩽ n. This result extends a theorem of Alspach (Canad. Math. Bull. 10 (1967) 283–286) for regular tournaments to regular multipartite tournaments.