Chromatic roots and minor-closed families of graphs

TL;DR

This paper investigates the smallest non-trivial roots of chromatic polynomials within minor-closed graph classes, providing exact values for three classes and conjecturing conditions for larger roots compared to the known universal bound.

Contribution

It determines the exact infimum of chromatic roots for three specific minor-closed graph classes and proposes a conjecture for when these roots exceed the universal bound of 32/27.

Findings

Exact infimum of chromatic roots for three minor-closed classes

Conjecture on conditions for roots exceeding 32/27

Extension of known bounds for chromatic roots

Abstract

Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G \in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32/27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32/27$.