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

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 de6ned 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)6 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 in6nite 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 su;cient condition for an almost regular 3-partite tournament D with the property that a given arc is contained in a Hamiltonian path of D. c © 2004 Elsevier B.V. All rights reserved.