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

We present an analysis of the structure and properties of chromatic polynomials P(G _pt,→m, q) of one-parameter and multi-parameter families of planar triangulation graphs G _pt,→m, where →m = (m1,..., m _p) is a vector of integer parameters. We use these to study the ratio of |P(G _pt,→m, τ+1)| to the Tutte upper bound (τ - 1) ^n-5, where τ = (1 + √5 )/2 and n is the number of vertices in G _pt,→m. 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 G _pt,→m with p = 1 and p = 2 and show that these have a structure of the form P(G _pt,m, q) = cG _pt,1λ ^m ₁ + cG _pt,2λ λ ^m ₂ + cG _pt,3λ λ ^m ₃ for p = 1, where λ ₁ = q - 2, λ ₂ = q - 3, and λ ₃ = -1, and P(G _pt,→m , q) = ∑ ³ _i1=1 ∑ ³ _i2=1 cG _{pt i1i2}, λ ^m1 _i1 λ ^m2 _i2 for p = 2. We derive properties of the coefficients cG _pt→i, and show that P(G _pt, →m , q) has a real chromatic zero that approaches (1/2)(3 + √ 5 ) 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.