Hamiltonian cycles in Dirac graphs


We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$$ exp_2 [2h(G) - n\log e - o(n)], $$

where h(G)=maxΣ e x e log(1/x e ), the maximum over x: E → ℜ+ satisfying Σ eυ x e = 1 for each υV, and log =log2. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1)))n. In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1))n, confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.

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


  1. [1]

    B. Bollobás: Martingales, isoperimetric inequalities and random graphs, in: Combinatorics (A. Hajnal, L. Lovász and V. T. Sós,eds.), Colloq. Math. Soc. János Bolyai 52, North Holland, 1988.

  2. [2]

    B. Bollobás: Extremal graph theory, in: Handbook of Combinatorics (R. L. Graham, M. Grötschel, L. Lovász, eds.), pp. 1231–1292, Elsevier Science, Amsterdam, 1995.

    Google Scholar 

  3. [3]

    J. A. Bondy: Basic graph theory, in: Handbook of Combinatorics (R. L. Graham, M. Grötschel, L. Lovász, eds.), pp. 3–110, Elsevier Science, Amsterdam, 1995.

    Google Scholar 

  4. [4]

    L. Brégman: Some properties of nonnegative matrices and their permanents, Math. Doklady 14 (1973), 945–949.

    MATH  Google Scholar 

  5. [5]

    I. Csiszár and J. Körner: Information theory. Coding theorems for discrete memoryless systems; Akadémiai Kiadó, Budapest, 1981.

    MATH  Google Scholar 

  6. [6]

    B. Cuckler: Hamiltonian cycles in regular tournaments, Combinatorics, Probability and Computing 16(2) (2007), 239–249.

    Article  MathSciNet  Google Scholar 

  7. [7]

    B. Cuckler and J. Kahn: Entropy bounds for perfect matchings and Hamiltonian cycles, Combinatorica 29(3) (2009), 327–335.

    Article  Google Scholar 

  8. [8]

    G. A. Dirac: Some theorems on abstract graphs, Proceedings of the London Mathematics Society, Third Series 2 (1952), 69–81.

    Article  MathSciNet  Google Scholar 

  9. [9]

    R. Durrett: Probability: Theory and Examples; Wadsworth, Belmont, 1991.

    MATH  Google Scholar 

  10. [10]

    M. Jerrum, A. Sinclair and E. Vigoda: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries, J. ACM 51 (2004), 671–697.

    Article  MathSciNet  Google Scholar 

  11. [11]

    J. Kahn and J. H. Kim: Random matchings in regular graphs, Combinatorica 18(2) (1998), 201–226.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    N. Linial, A. Samorodnitsky and A. Wigderson: A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents, Combinatorica 20(4) (2000), 545–568.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    R. J. McEliece: The theory of information and coding, Addison-Wesley, London, 1977.

    MATH  Google Scholar 

  14. [14]

    M. Molloy and B. Reed: Graph colouring and the probabilistic method, Springer, Berlin, 2002.

    MATH  Google Scholar 

  15. [15]

    J. Moon and L. Moser: On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963), 163–165.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    C. Nash-Williams: Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency, in: Studies in Pure Mathematics (L. Mirsky, ed.), pp. 157–183, Academic Press, London, 1971.

    Google Scholar 

  17. [17]

    O. Ore: Hamilton connected graphs, J. Math. Pures Appl. (9) 42 (1963), 21–27.

    MATH  MathSciNet  Google Scholar 

  18. [18]

    V. Rödl: On a packing and covering problem, Europ. J. Combinatorics 6 (1985), 69–78.

    MATH  Google Scholar 

  19. [19]

    G. Sárközy, S. Selkow and E. Szemerédi: On the number of Hamiltonian cycles in Dirac graphs, Discrete Mathematics 265 (2003), 237–250.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    E. Szemerédi: Regular partitions of graphs, in: Problèmes Combinatoires et Théorie des Graphes, Orsay (1976), Colloques Internationaux CNRS 260, pp. 399–401.

Download references

Author information



Corresponding author

Correspondence to Bill Cuckler.

Additional information

Supported by NSF grant DMS0200856.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cuckler, B., Kahn, J. Hamiltonian cycles in Dirac graphs. Combinatorica 29, 299–326 (2009). https://doi.org/10.1007/s00493-009-2360-2

Download citation

Mathematics Subject Classification (2000)

  • 05A16
  • 05C38
  • 05C45
  • 05D40
  • 05C70