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We derive improved upper bounds on the number of crossing-free straight-edgespanning cycles (also known as Hamiltonian tours and simple polygonizations)that can be embedded over any specific set of $N$ points in the plane. Morespecifically, we bound the ratio between the number of spanning cycles (orperfect matchings) that can be embedded over a point set and the number oftriangulations 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 newupper bound of $O(54.543^N)$ on the number of crossing-free straight-edgespanning cycles that can be embedded over any specific set of $N$ points in theplane (improving upon the previous best upper bound $O(68.664^N)$). Ouranalysis is based on Kasteleyn's linear algebra technique.

Counting Plane Graphs: Perfect Matchings, Spanning Cycles, and Kasteleyn's Technique