Skip to main content
    • Aa
    • Aa

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *