On cycles through a given vertex in multipartite tournaments Original Research Article

An n-partite tournament is an orientation of a complete n-partite graph, and an m-cycle is a directed cycle of length m. If D is a strongly connected n-partite tournament with the partite sets V1, V2, …, Vn and v an arbitrary vertex of D, then we shall prove the following statements. • • The vertex v is contained in a longest cycle. • • If n ≥ 3, then v belongs to an m-cycle or (m + 1)-cycle for all m ϵ {3,4, …, n}. • • If n ⩾ 4 and |Vi| ⩾ 2 for i = 1,2, …, n, then v belongs to an (n + 1)-cycle or (n + 2)-cycle. As easy consequences, we obtain known results of Bondy (1976), Ayel (1981), and Gutin (1984).