On the number of crossing-free matchings, (cycles, and partitions)

Abstract

We show that a set of n points in the plane has at most O(10.05n) perfect matchings with crossing-free straight-line embedding. The expected number of perfect crossing-free matchings of a set of n points drawn i.i.d. from an arbitrary distribution in the plane is at most O(9.24n).Several related bounds are derived: (a) The number of all (not necessarily perfect) crossing-free matchings is at most O(10.43n). (b) The number of left-right perfect crossing-free matchings (where the points are designated as left or as right endpoints of the matching edges) is at most O(5.38n). (c) The number of perfect crossing-free matchings across a line (where all the matching edges must cross a fixed halving line of the set) is at most 4n.These bounds are employed to infer that a set of n points in the plane has at most O(86.81n) crossing-free spanning cycles (simple polygonizations), and at most O(12.24n) crossing-free partitions (partitions of the point set, so that the convex hulls of the individual parts are pairwise disjoint).