A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees

TL;DR

This paper establishes a zero-free interval for the chromatic polynomial of graphs with a spanning tree of at most three leaves, identifying a specific root boundary and constructing graphs with roots approaching this boundary.

Contribution

It introduces a new zero-free interval for chromatic polynomials of such graphs and uses Whitney 2-switches to analyze an infinite class of these polynomials.

Findings

Chromatic polynomials of graphs with 3-leaf spanning trees have no roots in (1, t_1].

Constructed graphs with roots approaching t_1 from above.

Identified t_1 as the smallest real root of a specific polynomial.

Abstract

It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.