Counting colorings of triangle-free graphs

TL;DR

This paper establishes near-optimal lower bounds on the number of proper colorings of triangle-free graphs with given maximum degree, improving previous bounds and providing new proof techniques based on Rosenfeld's counting method.

Contribution

It proves a tight lower bound on the number of colorings for triangle-free graphs, and offers an alternative proof of Molloy's chromatic number bound using Rosenfeld's counting method.

Findings

Lower bound on colorings matches random regular graphs.

Improves previous bounds on the number of colorings.

Provides an alternative proof of Molloy's chromatic bound.

Abstract

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).