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

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.

arXiv comment16 pages, 5 figures