The Cycles In Regular Multipartite Tournaments

An important research direction for digraphs is about their cycles. Multipartite tournaments, as the generalization of tournaments, is an important class of digraphs. Therefore the investigation on cycles in multipartite tournaments has become intensive and productive. The paper is just about it.The paper consists of five chapters. In chapter1, we introduce some general concepts in this paper. In chapter2, we review some known results on multipartite tournaments. In chapter3, we prove that if D is an r-regular3-partite tournament with r≥2, then every vertex of D is contained in a6-cycle and every are of D is contained in a5-or6-cycle. In chapter4, we prove that each vertex of a regular c-partite tournament with c>16is contained in a strong subtournament of order p for every p€{3,4,...,c}. In chapter5, we prove the following result. Let A1, A2,..., Ap, B1, B2,..., Bq be some sets satisfying the following conditions:(a).|A|≥|A2|≥...≥|Ap|≥0; p g(b). UA∈U Bi; i=1i=1(c). For every i∈{1,2,..., q}, Bi is an m-set and Bi does not contain distinct elements x, y such that x∈Aj, y∈A^and j≠k, then we give out the lower bound of q. And the result is very useful for proving many results about cycles in strong multipartite tournaments.