On derivatives and higher-order derivatives of chromatic polynomials

TL;DR

This paper studies properties of chromatic polynomial derivatives, improving bounds for inequalities and confirming conjectures about their monotonicity and negativity in specific ranges.

Contribution

It advances understanding of chromatic polynomial derivatives by proving conjectures and improving bounds for their monotonicity and negativity.

Findings

Proved the inequality for all real x ≥ 10Δ^{3/2}.

Confirmed the conjecture that derivatives of log of chromatic polynomial are negative for x ≤ -3.01Δk.

Improved the bound for Dong's inequality from x ≥ 36Δ^{3/2} to x ≥ 10Δ^{3/2}.

Abstract

Let \( G \) be a graph of order \( n \) with maximum degree $\Delta$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)/x^n$. Dong proved that $(x-1)^nP(G,x)\geq x^nP(G,x-1)$ for all real $x\geq n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)/P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\geq 36\Delta^{3/2}$. We improve this bound by proving that it also holds for all real $x\geq 10\Delta^{3/2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that \( \frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0 \) for all \( k \geq 2 \) and \( x \in (-\infty, 0) \). We establish this conjecture for all \( k \geq 2 \) and \( x\leq -3.01\Delta k \).