Packing Plane Perfect Matchings into a Point Set

TL;DR

This paper investigates the maximum number of plane perfect matchings that can be packed into a set of points in the plane, providing lower bounds and exact counts for special configurations.

Contribution

It establishes a logarithmic lower bound on the number of plane perfect matchings that can be packed into any point set and determines exact counts for certain special cases.

Findings

At least loor(log_2 n) plane perfect matchings can be packed into any point set.

Exact number of matchings is determined for some special point configurations.

Extensions of the problem are also discussed.

Abstract

Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? We prove that at least $\lceil\log_2{n}\rceil-2$ plane perfect matchings can be packed into any point set $P$. For some special configurations of point sets, we give the exact answer. We also consider some extensions of this problem.