Three supplements to Reid’s theorem in multipartite tournaments

Abstract

An n -partite tournament is an orientation of a complete n -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 D be an ( α ( D ) + 1 )-strong n -partite tournament with n ≥ 6 , where α ( D ) is the independence number of D , then D contains two disjoint cycles of lengths 3 and n − 3 , respectively, unless D is isomorphic to the 7-tournament containing no transitive 4-subtournament (denoted by T 7 1 ). The second one is obtained by considering the number of partite sets that cycles pass through: every ( α ( D ) + 1 )-strong n -partite tournament D with n ≥ 6 contains two disjoint cycles which contain vertices from exactly 3 and n − 3 partite sets, respectively, unless it is isomorphic to T 7 1 . The last one is about two disjoint cycles passing through all partite sets.