The number of non-crossing perfect plane matchings is minimized (almost) only by point sets in convex position

TL;DR

This paper proves that the minimum number of non-crossing perfect matchings occurs only when points are in convex position, with one exception for six points, highlighting the uniqueness of convex configurations.

Contribution

It establishes that convex position uniquely minimizes non-crossing perfect matchings, except for a specific six-point case, extending previous lower bound results.

Findings

Convex point sets uniquely minimize non-crossing perfect matchings.

An exceptional six-point configuration also attains the minimum.

The minimum number of matchings is the Catalan number C_k.

Abstract

It is well-known that the number of non-crossing perfect matchings of $2k$ points in convex position in the plane is $C_k$, the $k$th Catalan number. Garc\'ia, Noy, and Tejel proved in 2000 that for any set of $2k$ points in general position, the number of such matchings is at least $C_k$. We show that the equality holds only for sets of points in convex position, and for one exceptional configuration of $6$ points.