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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Ferber, Asaf Krivelevich, Michael and Sudakov, Benny 2016. Counting and packing Hamilton cycles in dense graphs and oriented graphs. Journal of Combinatorial Theory, Series B,

    Ferber, Asaf Kronenberg, Gal and Long, Eoin 2015. Packing, Counting and Covering Hamilton cycles in random directed graphs. Electronic Notes in Discrete Mathematics, Vol. 49, p. 813.

    Kühn, Daniela and Osthus, Deryk 2012. A survey on Hamilton cycles in directed graphs. European Journal of Combinatorics, Vol. 33, Issue. 5, p. 750.

    Cuckler, Bill and Kahn, Jeff 2009. Hamiltonian cycles in Dirac graphs. Combinatorica, Vol. 29, Issue. 3, p. 299.

  • Combinatorics, Probability and Computing, Volume 16, Issue 2
  • March 2007, pp. 239-249

Hamiltonian Cycles in Regular Tournaments

  • DOI:
  • Published online: 01 March 2007

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(n0.25n!/2n) for arbitrary tournaments due to Friedgut and Kahn (somewhat improving Alon's bound of O(n0.5n!/2n)). A key ingredient of the proof is a martingale analysis of self-avoiding walks on a regular tournament.

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[1]I. Adler , N. Alon and S. M. Ross (2001) On the maximum number of Hamiltonian paths in tournaments. Random Struct. Alg. 18 291296.

[2]N. Alon (1990) The maximum number of Hamiltonian paths in tournaments. Combinatorica 10 319324.

[3]N. Alon and J. Spencer (2000) The Probabilistic Method, 2nd edn, Wiley-Interscience, New York.

[7]E. Friedgut and J. Kahn (2005) On the number of Hamiltonian cycles in a tournament. Combin. Probab. Comput. 14 769781.

[8]G. Grimmett (1999) Percolation, 2nd edn, Springer, New York.

[11]C. J. H. McDiarmid (1989) On the method of bounded differences. In Surveys in Combinatorics 1989: Invited Papers at the 12th British Combinatorial Conference (J. Siemons , ed.), Cambridge University Press, pp. 148188.

[13]J. Radhakrishnan (1997) An entropy proof of Brégman's Theorem. J. Combin. Theory Ser. A 77 161164.

[15]A. Schrijver (1978) A short proof of Minc's conjecture. J. Combin. Theory Ser. A 25 8083.

[18]C. Thomassen (1980) Hamiltonian-connected tournaments J. Combin. Theory Ser. B 28 142163.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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