Almost regular multipartite tournaments containing a Hamiltonian path through a given arc

Abstract

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d +( x) and d −( x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by i g ( D)=max{ d +( x), d −( x)}−min{ d +( y), d −( y)} over all vertices x and y of D (including x= y). If i g ( D)=0, then D is regular and if i g ( D)⩽1, then D is called almost regular. Recently, Volkmann and Yeo have proved that every arc of a regular multipartite tournament is contained in a directed Hamiltonian path. If c⩾4, then this result remains true for almost all c-partite tournaments D of a given constant irregularity i g ( D). For the case that i g ( D)=1 we will give a more detailed analysis. If c=3, then there exist infinite families of such digraphs, which have an arc that is not contained in a directed Hamiltonian path of D. Nevertheless, we will present an interesting sufficient condition for an almost regular 3-partite tournament D with the property that a given arc is contained in a Hamiltonian path of D.