Hamiltonian paths, containing a given path or collection of arcs, in close to regular multipartite 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)=0, then D is called regular. Let V1,V2,…,Vc be the partite sets of a c-partite tournament D such that |V1|⩽|V2|⩽⋯⩽|Vc|. If P is a directed path of length q in the c-partite tournament D such that|V(D)|⩾2ig(D)+3q+2|Vc|+|Vc−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