Hamiltonian paths containing a given arc, in almost regular bipartite tournaments

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 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)⩽1, then D is called almost regular, and if i g( D)=0, then D is regular. More than 10 years ago, Amar and Manoussakis and independently Wang proved that every arc of a regular bipartite tournament is contained in a directed Hamiltonian cycle. In this paper, we prove that every arc of an almost regular bipartite tournament T is contained in a directed Hamiltonian path if and only if the cardinalities of the partite sets differ by at most one and T is not isomorphic to T 3,3, where T 3,3 is an almost regular bipartite tournament with three vertices in each partite set. As an application of this theorem and other results, we show that every arc of an almost regular c-partite tournament D with the partite sets V 1, V 2,…, V c such that | V 1|=| V 2|=⋯=| V c |, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T 3,3.