Combinatorica

, Volume 35, Issue 2, pp 127–151 | Cite as

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

Original Paper

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.

Mathematics Subject Classification (2000)

05C31 05C60 05C15 82B20 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  2. 2.Department of Computer ScienceEötvös Loránd UniversityBudapestHungary

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