Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

TL;DR

This paper studies the convergence of chromatic measures in sparse graphs and shows that for Benjamini-Schramm convergent sequences, these measures and related polynomial properties converge analytically, extending previous results.

Contribution

It establishes the convergence of chromatic measures and the normalized log of chromatic polynomials for Benjamini-Schramm convergent graph sequences, generalizing prior work.

Findings

Chromatic measures converge in holomorphic moments for convergent graph sequences.

Normalized log of chromatic polynomial converges to an analytic function outside a bounded disk.

Provides explicit estimates on proper colorings for graphs with large girth.

Abstract

We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. As a corollary, for a convergent sequence of finite graphs, we prove that the normalized log of the chromatic polynomial converges to an analytic function outside a bounded disc. This generalizes a recent result of Borgs, Chayes, Kahn and Lov\'asz, who proved convergence at large enough positive integers and answers a question of Borgs. Our methods also lead to explicit estimates on the number of proper colorings of graphs with large girth.