Almost all almost regular c-partite tournaments with c⩾5 are vertex pancyclic

Abstract

A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d +( x) be the outdegree and d −( x) the indegree of the vertex x in D. The minimum (maximum) outdegree and the minimum (maximum) indegree of D are denoted by δ + ( Δ +) and δ − ( Δ −), respectively. In addition, we define δ=min{ δ +, δ −} and Δ=max{ Δ +, Δ −}. A digraph is regular when δ= Δ and almost regular when Δ− δ⩽1. Recently, the third author proved that all regular c-partite tournaments are vertex pancyclic when c⩾5, and that all, except possibly a finite number, regular 4-partite tournaments are vertex pancyclic. Clearly, in a regular multipartite tournament, each partite set has the same cardinality. As a supplement of Yeo's result we prove first that an almost regular c-partite tournament with c⩾5 is vertex pancyclic, if all partite sets have the same cardinality. Second, we show that all almost regular c-partite tournaments are vertex pancyclic when c⩾8, and third that all, except possibly a finite number, almost regular c-partite tournaments are vertex pancyclic when c⩾5.