Complementary cycles of all lengths in tournaments
Journal of Combinatorial Theory Series B
On cycles through a given vertex in multipartite tournaments
Selected papers from the second Krakow conference on Graph theory
Vertex deletion and cycles in multipartite tournaments
Discrete Mathematics
Extendable Cycles in Multipartite Tournaments
Graphs and Combinatorics
Digraphs: Theory, Algorithms and Applications
Digraphs: Theory, Algorithms and Applications
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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 (@a(D)+1)-strong n-partite tournament with n=6, where @a(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 (@a(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.