Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

  • Authors:

  • Alan D. Sokal

  • Affiliations:

  • Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA (e-mail: sokal@nyu.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2001

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Abstract

We show that there exist universal constants C(r) G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc ∣q∣ C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Potts-model partition function ZG(q, {ve}) in the complex antiferromagnetic regime ∣1 + ve∣ ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Kotecký–Preiss condition for nonvanishing of a polymer-model partition function. We also show that, for all loopless graphs G of second-largest degree ≤ r, the zeros of PG(q) lie in the disc ∣q∣ C(r) + 1. Along the way, I give a simple proof of a generalized (multivariate) Brown–Colbourn conjecture on the zeros of the reliability polynomial for the special case of series-parallel graphs.