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

TL;DR

This paper proves a conjecture about the negativity of derivatives of the logarithm of the chromatic polynomial with negative variables, specifically for all derivatives of order two or higher under certain conditions related to the graph's maximum degree.

Contribution

The paper confirms a conjecture regarding the properties of chromatic polynomials with negative variables for a broad class of graphs, extending previous partial results.

Findings

The conjecture holds for all derivatives of order ≥ 2 under specified conditions.

The result applies to all graphs with maximum degree Δ, for x ≤ -6.66Δk.

Provides bounds for the validity of the conjecture based on graph degree and derivative order.

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 $G$.