Counting colorings of triangle-free graphs
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By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has chromatic number at most (C+o(1))Δ/logΔ for some universal constant C > 0. Using the entropy compression method, Molloy proved that one can in fact take C = 1. Here we show that for every q ≥ (1 + o(1))Δ/logΔ, the number c(G,q) of proper q-colorings of G satisfies c(G, q) ≥ (1 - 1/q)^m ((1-o(1))q)^n, where n = |V(G)| and m = |E(G)|. Except for the o(1) term, this lower bound is best possible as witnessed by random Δ-regular graphs. When q = (1 + o(1)) Δ/logΔ, our result yields the inequality c(G,q) ≥ exp((1 - o(1)) logΔ/2 n), which implies the optimal lower bound on the number of independent sets in G due to Davies, Jenssen, Perkins, and Roberts. An important ingredient in our proof is the counting method that was recently developed by Rosenfeld. As a byproduct, we obtain an alternative proof of Molloy's bound χ(G) ≤ (1 + o(1))Δ/logΔ using Rosenfeld's method in place of entropy compression.
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