The structure of chromatic polynomials of planar triangulations and implications for chromatic zeros and asymptotic limiting quantities

Abstract

We present an analysis of the structure and properties of chromatic polynomials of one-parameter and multi-parameter families of planar triangulation graphs , where is a vector of integer parameters. We use these to study the ratio of to the Tutte upper bound (τ − 1)n − 5, where and n is the number of vertices in . In particular, we calculate limiting values of this ratio as n → ∞ for various families of planar triangulations. We also use our calculations to analyze zeros of these chromatic polynomials. We study a large class of families with p = 1 and p = 2 and show that these have a structure of the form for p = 1, where λ1 = q − 2, λ2 = q − 3, and λ3 = −1, and for p = 2. We derive properties of the coefficients and show that has a real chromatic zero that approaches as one or more of the mi → ∞. The generalization to p ⩾ 3 is given. Further, we present a one-parameter family of planar triangulations with real zeros that approach 3 from below as m → ∞. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.