Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn's technique

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(1.8181N) for cycles and O(1.1067N) for matchings. These imply a new upper bound of O(54.543N) 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(68.664N)). Our analysis is based on a weighted variant of Kasteleyn's linear algebra technique.

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