On Brenti's conjecture about the log-concavity of the chromatic polynomial

TL;DR

This paper proves a modified version of Brenti's conjecture, showing that the chromatic polynomial of a graph is log-concave for sufficiently large values related to the graph's maximum degree, using root location bounds.

Contribution

The authors establish a new bound for the log-concavity of the chromatic polynomial, improving previous conjectures by relating it to the graph's maximum degree and root location results.

Findings

Chromatic polynomial is log-concave for all q > CΔ + 1 with explicit C < 10.

Counterexample shows the bound does not hold for C < 1.

Uses root bounds from Sokal and Borgs to prove log-concavity.

Abstract

The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these polynomials has also been well studied. One famous result due to A. Sokal and C. Borgs provides a bound on the absolute value of the roots of the chromatic polynomial in terms of the highest degree of the graph. We use this result to prove a modification of a log-concavity conjecture due to F. Brenti. The original conjecture of Brenti was that the chromatic polynomial is log-concave on the natural numbers. This was disproved by Paul Seymour by presenting a counter example. We show that the chromatic polynomial $P_G(q)$ of graph $G$ is in fact log-concave for all $q > C\Delta + 1$ for an explicit constant $C < 10$, where $\Delta$ denotes the highest degree of $G$. We also provide an example which shows that the result is not true for constants $C$ smaller than 1.