Lower bounds on the number of crossing-free subgraphs of KN
Computational Geometry: Theory and Applications
Computational Geometry in C
A better upper bound on the number of triangulations of a planar point set
Journal of Combinatorial Theory Series A
On the Number of Crossing-Free Matchings, Cycles, and Partitions
SIAM Journal on Computing
On the Number of Plane Geometric Graphs
Graphs and Combinatorics
Fast enumeration algorithms for non-crossing geometric graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
On degrees in random triangulations of point sets
Journal of Combinatorial Theory Series A
Counting plane graphs with exponential speed-up
Rainbow of computer science
On the number of cycles in planar graphs
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
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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 Kasteleyn@?s linear algebra technique.