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

Tutte proved that if G _pt is a planar triangulation and P(G _pt, q) is its chromatic polynomial, then |P(G _pt, τ+ 1)| ≤ (τ-1) ^n-5, where τ=(1+√5)/2 and n is the number of vertices in G _pt. Here we study the ratio r(G _pt) = |P(G _pt, τ+1)|/(τ-1) ^n-5 for a variety of planar triangulations. We construct infinite recursive families of planar triangulations G _{pt, m} depending on a parameter m linearly related to n and show that if P(G _{pt, m}, q) only involves a single power of a polynomial, then r(G _{pt, m}) approaches zero exponentially fast as n→∞. We also construct infinite recursive families for which P(G _{pt, m}, q) is a sum of powers of certain functions and show that for these, r(G _{pt, 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.