Hamiltonian paths, containing a given path or collection of arcs, in close to regular multipartite tournaments

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 V₁,V₂,...,V_c be the partite sets of a c-partite tournament D such that |V₁|≤|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.