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Hamiltonian Cycles in Regular Tournaments


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
  • URL: /core/journals/combinatorics-probability-and-computing
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