Hamiltonian paths, containing a given path or collection of arcs, in close to regular multipartite 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)=0, then D is called regular. Let V 1, V 2,…, V c be the partite sets of a c-partite tournament D such that | V 1|⩽| V 2|⩽⋯⩽| V c |. If P is a directed path of length q in the c-partite tournament D such that |V(D)|⩾2i g(D)+3q+2|V c|+|V c−1|−2, then we prove in this paper that there exists a Hamiltonian path in D, starting with the path P. Examples will show that this condition is best possible. As an application of this theorem, we prove that each arc of a regular multipartite tournament is contained in a Hamiltonian path. Some related results are also presented.