An entropy proof of Bergman's theorem
Journal of Combinatorial Theory Series A
On the maximum number of Hamiltonian paths in tournaments
Random Structures & Algorithms
On the Number of Hamiltonian Cycles in a Tournament
Combinatorics, Probability and Computing
A survey on Hamilton cycles in directed graphs
European Journal of Combinatorics
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We show that every regular tournament on $n$ vertices has at least $n!/(2+o(1))^n$ 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 $O(n^{0.25} n!/2^{n})$ for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon's bound of $O(n^{0.5} n!/2^{n})$). A key ingredient of the proof is a martingale analysis of self-avoiding walks on a regular tournament.