Entropy bounds for perfect matchings and Hamiltonian cycles

Abstract

For a graph G = (V,E) and x: E → ℜ⁺ satisfying Σ _e∋υ x _e = 1 for each υ ∈ V, 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(e∈f),

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

(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

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

.