We show that there exist universal constants C(r) < ∞ such that, for all loopless graphs
G of maximum degree [les ] r, the zeros (real or complex) of the chromatic polynomial
PG(q) lie in the disc [mid ]q[mid ] < C(r).
Furthermore, C(r) [les ] 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 [mid ]1 + ve[mid ] [les ] 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 [les ] r, the zeros of PG(q) lie in the disc
[mid ]q[mid ] < 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.
Potts Model on Infinite Graphs and the Limit of Chromatic Polynomials
