Entropy bounds for perfect matchings and Hamiltonian cycles


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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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