  We derive improved upper bounds on the number of crossing-free straight-edge
spanning cycles (also known as Hamiltonian tours and simple polygonizations)
that can be embedded over any specific set of $N$ points in the plane. More
specifically, we bound the ratio between the number of spanning cycles (or
perfect matchings) that can be embedded over a point set and the number of
triangulations that can be embedded over it. The respective bounds are
$O(1.8181^N)$ for cycles and $O(1.1067^N)$ for matchings. These imply a new
upper bound of $O(54.543^N)$ on the number of crossing-free straight-edge
spanning cycles that can be embedded over any specific set of $N$ points in the
plane (improving upon the previous best upper bound $O(68.664^N)$). Our
analysis is based on Kasteleyn's linear algebra technique.
