Improved bounds for the zeros of the chromatic polynomial via Whitney's Broken Circuit Theorem

TL;DR

This work proves new bounds on the zeros of chromatic polynomials for graphs of bounded degree by translating the problem through Whitney's Broken Circuit Theorem into a zero-free region for a forest-type generating function. The authors establish a strong zero-free disc of radius $5.94\,\Delta$ for general graphs and derive improved girth-dependent bounds $K_g$ whose limit is at most $3.86$ as $g \to \infty$, via careful inductive estimates on tree-related generating functions and broken-circuit-free structures. A second major contribution is a zero-free disc for the forest generating function $Z_G(x)$, showing $|x| \le 1/(2\Delta)$ suffices, with extensions to multigraphs and path-based decompositions. The results have implications for approximate counting of colorings, phase-transition analysis in the arboreal gas, and related combinatorial-probabilistic models, and they open avenues for further tightening of constants and for multivariate/generalized settings.

Abstract

We prove that for any graph $G$ of maximum degree at most $Δ$, the zeros of its chromatic polynomial $χ_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 Δ$ centered at $0$. This improves on the previously best known bound of approximately $6.91Δ$. We also obtain improved bounds for graphs of high girth. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $Δ$ and girth at least $g$, the zeros of its chromatic polynomial $χ_G(x)$ lie inside the disc of radius $K_g Δ$ centered at $0$, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g < 5$ when $g \geq 5$ and $K_g < 4$ when $g \geq 25$ and $K_g$ tends to approximately $3.86$ as $g \to \infty$. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph $G$ to the generating function of so-called broken-circuit-free forests in $G$. We also establish a zero-free disc for the generating function of all forests in $G$ (aka the partition function of the arboreal gas) which may be of independent interest.

Figures (1)

• Figure 1: Values for $a$ and $b$ and $g$ for which $f_g(a,b)<a$ and corresponding values for $h(a,b)^{-1}$. The rightmost column gives concrete bounds on the value of $K_g$ in the statement of Theorem \ref{['thm:large girth']}. For example, $K_5\leq 4.87264$ and $K_{25}\leq 3.97497$.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Whitney Whitney
• proof : Proof of Theorem \ref{['thm:whitney']}
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof