The maximum number of Hamiltonian paths in tournaments


Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n 3/2 · n!/2n−1, wherec is a positive constant independent ofn.

This is a preview of subscription content, access via your institution.


  1. [1]

    L. M. Brégman: Some properties of nonnegative matrices and their permanents,Soviet Math. Dokl. 14 (1973), 945–949 [Dokl. Akad. Nauk SSSR 211 (1973), 27–30].

    Google Scholar 

  2. [2]

    G. P. Egorichev: The solution of the Van der Waerden problem for permanents,Dokl. Akad. Nauk SSSR 258 (1981), 1041–1044.

    Google Scholar 

  3. [3]

    P. ErdŐs, andJ. H. Spencer:Probabilistic Methods in Combinatorics, Akadémiai Kiadó, Budapest,1974.

    Google Scholar 

  4. [4]

    D. I. Falikman: A proof of Van der Waerden's conjecture on the permanent of a doubly stochastic matrix,Mat. Zametki 19 (1981), 931–938.

    Google Scholar 

  5. [5]

    H. Minc: Upper bounds for permanents of (0,1)-matrices,Bull. Amer. Math. Soc. 69 (1963), 789–791.

    Google Scholar 

  6. [6]

    J. W. Moon:Topics on Tournaments, Holt, Reinhart and Winston, New York.1968.

    Google Scholar 

  7. [7]

    A. Schrijver: A short proof of Minc's Conjecture,J. Combinatorial Theory, Ser. A25 (1978), 80–83.

    Google Scholar 

  8. [8]

    J. H. Spencer:Ten Lectures on the Probabilistic Method, SIAM, Philadelphia,1987.

    Google Scholar 

  9. [9]

    T. Szele: Kombinatorikai vizsgálatok az irányított teljes gráffal kapcsolatban,Mat. Fiz. Lapok 50 (1943), 223–256. [For a German translation, see Kombinatorische Untersuchungen über gerichtete vollstÄndige Graphen,Publ. Math. Debrecen 13 (1966), 145–168.]

    Google Scholar 

Download references

Author information



Additional information

Research supported in part by a U.S.A.-Israel BSF grant and by a Bergmann Memorial Grant.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N. The maximum number of Hamiltonian paths in tournaments. Combinatorica 10, 319–324 (1990).

Download citation

AMS subject classification (1980)

  • 05 C 20
  • 05 C 35
  • 05 C 38