A Zero-Free Interval for Chromatic Polynomials of Nearly 3-Connected Plane Graphs

Let G = (V,E) be a 2-connected plane graph on n vertices with outer face C such that every 2-vertex cut of G contains at least one vertex of C. Let PG(q) denote the chromatic polynomial of G. We show that (−1)nPG(q) > 0 for all 1 < q ≤ 1.2040.... This result is a corollary of a more general result that (−1)ZG(q,w) > 0 for all 1 < q ≤ 1.2040..., where ZG(q,w) is the multivariate Tutte polynomial of G, w = {we}e∈E , we = −1 for all e which are not incident to a vertex of C, we ∈ W2 for all e ∈ E(C), we ∈ W1 for all other edges e, and W1,W2 are suitably chosen intervals with −1 ∈ W1 ⊂ W2 ⊆ (−2, 0).