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 martingale analysis of self-avoiding walks on a regular tournament.