Three supplements to Reid’s theorem in multipartite tournaments

An nn-partite tournament is an orientation of a complete nn-partite graph. In this paper, we give three supplements to Reid’s theorem [K.B. Reid, Two complementary circuits in two-connected tournaments, Ann. Discrete Math. 27 (1985) 321–334] in multipartite tournaments. The first one is concerned with the lengths of cycles and states as follows: let DD be an (α(D)+1α(D)+1)-strong nn-partite tournament with n≥6n≥6, where α(D)α(D) is the independence number of DD, then DD contains two disjoint cycles of lengths 3 and n−3n−3, respectively, unless DD is isomorphic to the 7-tournament containing no transitive 4-subtournament (denoted by T71). The second one is obtained by considering the number of partite sets that cycles pass through: every (α(D)+1α(D)+1)-strong nn-partite tournament DD with n≥6n≥6 contains two disjoint cycles which contain vertices from exactly 3 and n−3n−3 partite sets, respectively, unless it is isomorphic to T71. The last one is about two disjoint cycles passing through all partite sets.