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

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)1 and one of the partite sets of it consists of a single vertex, say v, then D 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,then Dhas 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.