Counting colorings of triangle-free graphs

By a theorem of Johansson, every triangle-free graph G maximum degree Δ has chromatic number at most (C+o(1))Δ/log⁡Δ for some universal constant C>0. Using the entropy compression method, Molloy proved that one can in fact take C=1. Here we show q⩾(1+o(1))Δ/log⁡Δ, c(G,q) proper q-colorings satisfiesc(G,q)⩾(1−1q)m((1−o(1))q)n, where n=|V(G)| and m=|E(G)|. Except o(1) term, this lower bound is best possible as witnessed by random Δ-regular graphs. When q=(1+o(1))Δ/log⁡Δ, our result yields inequalityc(G,q)⩾exp⁡((1−o(1))log⁡Δ2n), which improves an earlier Iliopoulos optimal value factor exponent. Furthermore, implies on independent sets due to Davies, Jenssen, Perkins, Roberts. An important ingredient proof counting method was recently developed Rosenfeld. As byproduct, obtain alternative Molloy's χ(G)⩽(1+o(1))Δ/log⁡Δ using Rosenfeld's place (other proofs technique were given independently Hurley Pirot Martinsson).