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) over right arrow), q) of one-parameter and multi-parameter families of planar triangulation graphs G(pt), ((m) over right arrow), where ((m) over right arrow) = (m(1), ... , m(p)) is a vector of integer parameters. We use these to study the ratio of vertical bar P(G(pt), ((m) over right arrow), tau + 1) vertical bar to the Tutte upper bound (tau - 1)(n-5), where tau = ( 1 + root 5)/2 and n is the number of vertices in G(pt), ((m) over right arrow). In particular, we calculate limiting values of this ratio as n -> infinity 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) over right arrow) with p = 1 and p = 2 and show that these have a structure of the form P(G(pt), (m), q) = (c)G(pt), 1 lambda(m)(1) + (c)G(pt), 2 lambda(m)(2) + (c)G(pt), 3 lambda(m)(3) for p = 1, where lambda(1) = q - 2, lambda(2) = q - 3, and lambda(3) = -1, and P(G(pt), ((m) over right arrow), q) = Sigma(3)(i1= 1) Sigma(3)(i2=1) cG(pt), i(1)i(2)lambda(m1)(i1) lambda(m2)(i2) for p = 2. We derive properties of the coefficients c(Gpt), ((i) over right arrow) and show that P(G(pt), ((m) over right arrow), q) has a real chromatic zero that approaches (1/2)(3 + v 5) as one or more of the m(i) -> infinity. 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 -> infinity. Implications for the ground-state entropy of the Potts antiferromagnet are discussed.