Point sets with many non-crossing matchings

TL;DR

This paper studies the maximum number of non-crossing perfect matchings in planar point sets, introduces a new method using down-free matchings, and improves lower bounds through novel geometric constructions and analysis.

Contribution

It introduces the concept of down-free matchings, simplifies existing bounds, and constructs new point sets with higher numbers of non-crossing matchings.

Findings

Double zigzag chain has approximately 3.0532^n matchings.

Generalized double r-chains achieve about 3.0930^n matchings.

New bounds improve previous lower bounds for non-crossing matchings.

Abstract

The maximum number of non-crossing straight-line perfect matchings that a set of $n$ points in the plane can have is known to be $O(10.0438^n)$ and $\Omega^*(3^n)$. The lower bound, due to Garc\'ia, Noy, and Tejel (2000) is attained by the double chain, which has $\Theta(3^n n^{O(1)})$ such matchings. We reprove this bound in a simplified way that uses the novel notion of down-free matching, and apply this approach on several other constructions. As a result, we improve the lower bound. First we show that double zigzag chain with $n$ points has $\Theta^*(\lambda^n)$ such matchings with $\lambda \approx 3.0532$. Next we analyze further generalizations of double zigzag chains - double $r$-chains. The best choice of parameters leads to a construction with $\Theta^*(\nu^n)$ matchings, with $\nu \approx 3.0930$. The derivation of this bound requires an analysis of a coupled dynamic-programming recursion between two infinite vectors.