Tutte proved that if $G_{ pt}$ is a planar triangulation and $P(G_{ pt},q)$ is its chromatic polynomial, then $|P(G_{ pt},\tau+1)| \le (\tau-1)^{ n-5}$, where $\tau=(1+\sqrt{ 5} \,)/2$ and $n$ is the number of vertices in $G_{ pt}$. Here we study the ratio $r(G_{ pt})=|P(G_{ pt},\tau+1)|/(\tau-1)^{ n-5}$ 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 $n \to \infty$. 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 $n \to \infty$. The connection between the Tutte upper bound and the observed chromatic zero(s) near to $\tau+1$ is investigated. We report the first known graph for which the zero(s) closest to $\tau+1$ is not real, but instead is a complex-conjugate pair. Finally, we discuss connections with nonzero ground-state entropy of the Potts antiferromagnet on these families of graphs.
Categories: math-ph math.CO math.MP