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

2025-05-07

AI Summary

This work establishes improved upper bounds on the moduli of complex zeros of the chromatic polynomial, thereby tightening the region containing these zeros in the complex plane. For general graphs, graphs of high girth, and claw-free graphs, we derive tighter disk-radius bounds—reducing the prior bound of $5.94Delta$ to $4.25Delta$, $3.60Delta$, and $3.81Delta$, respectively, where $Delta$ denotes the maximum degree. Our approach integrates tools from complex analysis, spectral graph theory, combinatorial enumeration, and recursive contraction–deletion techniques; notably, we introduce, for the first time, a characterization of chromatic polynomial coefficients via partially acyclic orientations. These advances significantly enhance the precision of zero localization and provide a more rigorous analytic foundation for studying phase transitions in graph coloring and related statistical physical models, such as the Potts model.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $chi_G(x)$, lie inside the disk centered at $0$ of radius $4.25 Delta(G)$, where $Delta(G)$ denote the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and the second author, who proved a bound of $5.94Delta(G)$. 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.

Problem

• Bounds on chromatic polynomial zeros for general graphs
• Improved radius for zeros in large-girth graphs
• Tighter bounds for claw-free graphs using orientations

Innovation

• Improved chromatic polynomial zero bounds
• Key novel acyclic orientations representation
• Enhanced disk radius for claw-free graphs