On the zero-free region for the chromatic polynomial of claw-free graphs with and without induced square and induced diamond

TL;DR

This paper establishes new bounds on the zero-free regions of the chromatic polynomial for claw-free graphs, especially those without induced squares or diamonds, based on a novel pair independence ratio parameter.

Contribution

It introduces the pair independence ratio as a key parameter and derives improved zero-free bounds for the chromatic polynomial in claw-free graphs, with sharper results for graphs without certain induced subgraphs.

Findings

Zeros of chromatic polynomial lie inside a specific disk depending on maximum degree and pair independence ratio.

Sharper bounds are obtained for graphs without induced squares and diamonds.

The results improve upon recent bounds by Bencs and Regts.

Abstract

Given a claw-free graph $G=(V,E)$ with maximum degree $\Delta$, we define the parameter $\kappa\in [0,1]$ as $\kappa={\max_{v\in V}|I_v|\over \lfloor\Delta^2/4\rfloor}$ where $I_v$ is the set of all independent pairs in the neighborhood of $v$. We refer to $\kappa$ as the pair independence ratio of $G$. We prove that for any claw-free graph $G$ with pair independence ratio at most $\kappa$ the zeros of its chromatic polynomial $P_G(q)$ lie inside the disk $D=\{q\in \mathbb{C}:~|q|< C_\kappa^0\Delta\}$, where $C_\kappa^0$ is an increasing function of $\kappa\in [0,1]$. If $G$ is also square-free and diamond free, the function $C_\kappa^0$ can be replaced by a sharper function $C_\kappa^1$. These bounds constitute an improvement upon results recently given by Bencs and Regts in ''Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs''.