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

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''.