By a theorem of Johansson, every triangle-free graph $G$ of maximum degree $\Delta$ has chromatic number at most $(C+o(1))\Delta/\log \Delta$ 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 $q \geq (1 + o(1))\Delta/\log \Delta$, the number $c(G,q)$ of proper $q$-colorings of $G$ satisfies $c(G, q) \,\geq\, \left(1 - \frac{1}{q}\right)^m ((1-o(1))q)^n$, where $n = |V(G)|$ and $m = |E(G)|$. Except for the $o(1)$ term, this lower bound is best possible as witnessed by random $\Delta$-regular graphs. When $q = (1 + o(1)) \Delta/\log \Delta$, our result yields the inequality $c(G,q) \,\geq\, \exp\left((1 - o(1)) \frac{\log \Delta}{2} n\right)$, which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result 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 $\chi(G) \leq (1 + o(1))\Delta/\log \Delta$ using Rosenfeld's method in place of entropy compression (other proofs of Molloy's theorem using Rosenfeld's technique were given independently by Hurley and Pirot and Martinsson).
翻译:根据约翰森的理论,每个无三角方块 $G$, 最大度为$Delta$, 最多为$(C+o(1))\Delta/\Delta$, 最多为$(C+o(1))\Delta/Delta$, 一些通用常数 $C > 0美元。 使用 entropy 压缩法, Mollo 证明, 事实上可以花1美元=1美元。 这里我们显示, 每1美元=Gq(1+o(1))\Delta/log Perelta$, 适当的美元(G,q)$(美元, q) 美元(美元) 美元(美元) 美元(美元) 美元(美元) 美元(美元) 美元(美元) 美元(美元) 美元(美元) 美元(美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元(美元) (美元) (美元(美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元(美元) (美元) (美元) (美元) (美元) (美元) (美元=(美元) (美元=(美元) (美元) (美元=(美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元) (美元=(美元) (美元) (美元) (美元) (美元) (美元) (