Entropy bounds for perfect matchings and Hamiltonian cycles

For a graph G = (V,E) and x : E → < satisfying ∑ e3v xe = 1 for each v ∈ V , set h(x) = ∑ e xe log(1/xe) (with log = log2). We show that for any n-vertex G, random (not necessarily uniform) perfect matching f satisfying a mild technical condition, and xe = Pr(e ∈ f), H(f) < h(x)− 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 [5], 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) := max ∑ e xe log(1/xe) (the maximum over x as above). For instance, for the number, Ψ(G), of Hamiltonian cycles in such a G, we have Ψ(G) = exp2[2h(G)− n log e− o(n)].