Chromatic Polynomials of Planar Triangulations, the Tutte Upper Bound, and Chromatic Zeros

TL;DR

This paper investigates the behavior of chromatic polynomials of planar triangulations relative to Tutte's upper bound, exploring their zeros, asymptotic ratios, and connections to Potts model entropy, revealing new complex zeros near $ au+1$.

Contribution

It introduces recursive families of planar triangulations with varying asymptotic ratios of chromatic polynomial evaluations, and reports the first known non-real chromatic zero near $ au+1$.

Findings

Ratios $r(G_{pt,m})$ approach zero exponentially for certain families.

Some families have ratios approaching a finite nonzero constant.

Discovered a graph with complex conjugate zeros closest to $ au+1$.

Abstract

Tutte proved that if $G_{pt}$ is a planar triangulation and $P(G_{pt},q)$ is its chromatic polynomial, then $|P(G_{pt},\tau+1)| \le (\tau-1)^{n-5}$, where $\tau=(1+\sqrt{5} \,)/2$ and $n$ is the number of vertices in $G_{pt}$. Here we study the ratio $r(G_{pt})=|P(G_{pt},\tau+1)|/(\tau-1)^{n-5}$ for a variety of planar triangulations. We construct infinite recursive families of planar triangulations $G_{pt,m}$ depending on a parameter $m$ linearly related to $n$ and show that if $P(G_{pt,m},q)$ only involves a single power of a polynomial, then $r(G_{pt,m})$ approaches zero exponentially fast as $n \to \infty$. We also construct infinite recursive families for which $P(G_{pt,m},q)$ is a sum of powers of certain functions and show that for these, $r(G_{pt,m})$ may approach a finite nonzero constant as $n \to \infty$. The connection between the Tutte upper bound and the observed chromatic zero(s) near to $\tau+1$ is investigated. We report the first known graph for which the zero(s) closest to $\tau+1$ is not real, but instead is a complex-conjugate pair. Finally, we discuss connections with nonzero ground-state entropy of the Potts antiferromagnet on these families of graphs.