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

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 Lovász, 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.

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Correspondence to Miklós Abért.

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Abért, M., Hubai, T. Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs. Combinatorica 35, 127–151 (2015). https://doi.org/10.1007/s00493-014-3066-7

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Mathematics Subject Classification (2000)

  • 05C31
  • 05C60
  • 05C15
  • 82B20