Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs

TL;DR

We study the zeros of the chromatic polynomial χ_G(x) for graphs with maximum degree $\Delta$ and prove all zeros lie in the disk |z| ≤ $4.25 \Delta(G)$. The bound is sharpened for special graph classes: |z| ≤ $3.60 \Delta(G)$ when the graph has large girth and |z| ≤ $3.81 \Delta(G)$ for claw-free graphs; asymptotically, for large girth, $K_{\Delta,\infty} \leq 1/W(e^{-1}) \approx 3.59$. The approach combines two coefficient-interpretations—Whitney’s broken-circuit-free forests and Greene–Zaslavsky acyclic orientations—with rooted-tree generating-function bounds and a polymer-inspired inductive scheme; a novel edge-ratio analysis is developed for claw-free graphs via star-forests. These results improve long-standing bounds, have algorithmic implications via Barvinok interpolation, and raise questions about optimal constants.

Abstract

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $χ_G(x)$, lie inside the disk centered at $0$ of radius $4.25 Δ(G)$, where $Δ(G)$ denote the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and the second author, who proved a bound of $5.94Δ(G)$. We moreover show that for graphs of sufficiently large girth we can replace $4.25$ by $3.60$ and for claw-free graphs we can replace $4.25$ by $3.81$. Our proofs build on the ideas developed by Jenssen, Patel and the second author, adding some new ideas. A key novel ingredient for claw-free graphs is to use a representation of the coefficients of the chromatic polynomial in terms of the number of certain partial acyclic orientations.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Whitney Whitney
• Lemma 2.2: Greene and Zaslavsky greene1983interpretation
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• proof : Proof of Lemma \ref{['lem:acyclic to chromatic']}