On a conjecture concerning the property of chromatic polynomials with negative variables

Abstract

Let $G$ be a graph of order $n$ and $P(G,x)$ be the chromatic polynomial of $G$. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that $\frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0$ holds for all $k \geq 2$ and $x \in (-\infty, 0)$. We prove this conjecture for all $k \geq 2 $ and $ x\leq -6.66\Delta k $, in which $\Delta$ is the maximum degree of $G$.