Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleynʼs 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 ( 1.8181 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 a weighted variant of Kasteleynʼs linear algebra technique.