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

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 $4.25\mathrm{\Delta }(G)$, where $\mathrm{\Delta }(G)$ denotes the maximum degree of $G$. This improves on a recent result of Jenssen, Patel and Regts, who proved a bound of $5.94\mathrm{\Delta }(G)$. 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.