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arXiv · 2505.04366

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

Ferenc Bencs·Guus Regts
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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 $\Delta(G)$ denotes the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and Regts, who proved a bound of $5.94\Delta(G)$. Moreover, we 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 add some substantially novel ideas to those developed by Jenssen, Patel, and Regts, while building on them. 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.

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Ferenc Bencs, Guus Regts. 2025-05-07. Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs. https://arxiv.org/abs/2505.04366

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