Chromatic polynomials of planar triangulations, the Tutte upper bound and chromatic zeros

Abstract

Tutte proved that if Gpt is a planar triangulation and P(Gpt, q) is its chromatic polynomial, then |P(Gpt, τ + 1)| ⩽ (τ − 1)n − 5, where and n is the number of vertices in Gpt. Here we study the ratio r(Gpt) = |P(Gpt, τ + 1)|/(τ − 1)n − 5 for a variety of planar triangulations. We construct infinite recursive families of planar triangulations Gpt, m depending on a parameter m linearly related to n and show that if P(Gpt, m, q) only involves a single power of a polynomial, then r(Gpt, m) approaches zero exponentially fast as n → ∞. We also construct infinite recursive families for which P(Gpt, m, q) is a sum of powers of certain functions and show that for these, r(Gpt, m) may approach a finite nonzero constant as n → ∞. The connection between the Tutte upper bound and the observed chromatic zero(s) near to τ + 1 is investigated. We report the first known graph for which the zero(s) closest to τ + 1 is not real, but instead is a complex-conjugate pair. Finally, we discuss connections with the nonzero ground-state entropy of the Potts antiferromagnet on these families of graphs.