  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.
