On the number of 7-cycles in regular n-tournaments

Hamiltonian Cycles in Regular Tournaments

We show that every regular tournament on n vertices has at least n!/(2+o(1)) Hamiltonian cycles, thus answering a question of Thomassen [17] and providing a partial answer to a question of Friedgut and Kahn [7]. This compares to an upper bound of about O(nn!/2) for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon’s bound of O(nn!/2)). A key ingredient of the proof is a ma...

On 4-cycles and 5-cycles in Regular Tournaments

First, some definitions. A tournament is regular of degree k if each point has indegree k and outdegree k: clearly such a tournament has 2k +1 points. The trivial tournament has just one point. A tournament T is doubly regular with subdegree t if it is non-trivial and any two points of T jointly dominate precisely t points; equivalently if T is non-trivial and for each point v of T, the subtour...

Complementary cycles in regular multipartite tournaments

On Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n ≥ 4, then each arc of T lies on a cycle whose vertices are from exactly k partite sets for k = 4, 5, . . . , n. Our result, in a sense, generalizes a theorem due to Alspach.

When n-cycles in n-partite tournaments are longest cycles

An n-tournament is an orientation of a complete n-partite graph. It was proved by J.A. Bondy in 1976 that every strongly connected n-partite tournament has an n-cycle. We characterize strongly connected n-partite tournaments in which a longest cycle is of length n and, thus, settle a problem in L. Volkmann, Discrete Math. 245 (2002) 19-53.