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 $\kappa =\frac{{\max }_{v\in V}|I_v|}{\lfloor {\Delta }^2/4\rfloor }$ 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''.