Entropy bounds for perfect matchings and Hamiltonian cycles

Abstract

For a graph G = (V,E) and x: E → ℜ+ satisfying Σ eυ x e = 1 for each υV, set h(x) = Σ e x e log(1/x e ) (with log = log2). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and x e =Pr(ef),

$$ H(f) < h(x) - \frac{n} {2}\log e + o(n) $$

(where H is binary entropy). This implies a similar bound for random Hamiltonian cycles.

Specializing these bounds completes a proof, begun in [6], of a quite precise determination of the numbers of perfect matchings and Hamiltonian cycles in Dirac graphs (graphs with minimum degree at least n/2) in terms of h(G):=maxΣ e x e log(1/x e ) (the maximum over x as above). For instance, for the number, Ψ(G), of Hamiltonian cycles in such a G, we have

$$ \Psi (G) = exp_2 [2h(G) - n\log e - o(n)]. $$

.

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References

  1. [1]

    N. Alon and S. Friedland: The maximum number of perfect machings in graphs with a given degree sequence, Elec. J. Combinatorics 15 (2008), #N13.

    Google Scholar 

  2. [2]

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

    MATH  Google Scholar 

  3. [3]

    F. R. K. Chung, P. Frankl, R. Graham and J. B. Shearer: Some intersection theorems for ordered sets and graphs, J. Combin. Theory Ser. A 48 (1986), 23–37.

    Article  MathSciNet  Google Scholar 

  4. [4]

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

    Google Scholar 

  5. [5]

    B. Cuckler: Hamiltonian cycles in regular tournaments and Dirac graphs, Ph.D. thesis, Rutgers University, 2006.

  6. [6]

    B. Cuckler and J. Kahn: Hamiltonian cycles in Dirac graphs, Combinatorica 29(3) (2009), 299–326.

    Article  Google Scholar 

  7. [7]

    G. A. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc., Third Series 2 (1952), 69–81.

    Article  MathSciNet  Google Scholar 

  8. [8]

    G. P. Egorychev: Permanents (Russian), Krasnoyarsk, SFU, 2007.

    Google Scholar 

  9. [9]

    J. Kahn: An entropy approach to the hard-core model on bipartite graphs, Combinatorics, Probability and Computing 10 (2001), 219–237.

    MATH  MathSciNet  Google Scholar 

  10. [10]

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

    Google Scholar 

  11. [11]

    J. Radhakrishnan: An entropy proof of Bregman’s Theorem, J. Combin. Theory Ser. A 77 (1997), 161–164.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

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

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    A. Schrijver: A short proof of Minc’s conjecture, J. Combin. Theory Ser. A 25(1) (1978), 80–83.

    MATH  Article  MathSciNet  Google Scholar 

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Correspondence to Bill Cuckler.

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Supported by NSF grant DMS0200856.

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Cuckler, B., Kahn, J. Entropy bounds for perfect matchings and Hamiltonian cycles. Combinatorica 29, 327–335 (2009). https://doi.org/10.1007/s00493-009-2366-9

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Mathematics Subject Classification (2000)

  • 05A16
  • 05C38
  • 05C45
  • 05C70
  • 94A17