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

‡ Abstract. Let G ¼ð V;EÞ be a nonseparable plane graph on n vertices with at least two edges. Suppose that G has outer face C and that every 2-vertex-cut of G contains at least one vertex of C. Let PGðqÞ denote the chromatic polynomial of G. We show that ð−1Þ n PGðqÞ > 0 for all 1 0 for all 1 < q ≤ 1.2040:::, where ZGðq;wÞ is the multivariate Tutte polynomial of G, w ¼f wege∈E, we ¼ −1 for all e which are not incident to a vertex of C, we ∈ W2 for all e ∈ EðC Þ, we ∈ W1 for all other edges e, and W1, W2 are suitably chosen intervals with −1 ∈ W1 ⊂ W2 ⊆ ð−2;0Þ.