Counting colorings of triangle-free graphs
Anton Bernshteyn, Tyler Brazelton, Ruijia Cao, Akum Kang

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 of maximum degree has chromatic number at most for some universal constant . Using the entropy compression method, Molloy proved that one can in fact take . Here we show that for every , the number of proper -colorings of satisfies , where and . Except for the term, this lower bound is best possible as witnessed by random -regular graphs. When , our result yields the inequality , which improves an earlier bound of Iliopoulos and yields the optimal value for the constant factor in the exponent. Furthermore, this result…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph theory and applications
