C-Pancyclic Partial Ordering and (c–1)-Pan-Outpath Partial Ordering in Semicomplete Multipartite Digraphs

Abstract

An outpath of a vertex v in a digraph is a path starting at v such that v dominates the end vertex of the path only if the end vertex also dominates v. First we show that letting D be a strongly connected semicomplete c-partite digraph (c ≥ 3), and one of the partite sets of it consists of a single vertex, say v ,t henD has a c-pancyclic partial ordering from v, which generalizes a result about pancyclicity of multipartite tournaments obtained by Gutin in 1993. Then we prove that letting D be a strongly connected semicomplete c-partite digraph with c ≥ 3 and letting v be a vertex of D ,t henD has a (c − 1)-pan-outpath partly ordering from v. This result improves a theorem about outpaths in semicomplete multipartite digraphs obtained by Guo in 1999.