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

AbstractA 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 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)⩽1, then D is called almost regular, and if ig(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 T3,3, where T3,3 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 V1,V2,…,Vc such that |V1|=|V2|=⋯=|Vc|, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T3,3