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 ig(D)=max{d+(x),d−(x)}−min{d+(y),d−(y)} over all vertices x and y of D (including x=y). If ig(D)=0, then D is regular and if ig(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 ig(D). For the case that ig(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.