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

Micha Sharir, Adam Sheffer, Emo Welzl
2013 Journal of combinatorial theory. Series A  
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
more » ... hings. 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 Kasteleyn's linear algebra technique.
doi:10.1016/j.jcta.2013.01.002 fatcat:u62vgjo7o5azthn7qbwjybgixm