Compatible Geometric Matchings

TL;DR

This paper explores the structure of non-crossing geometric perfect matchings, demonstrating that any two such matchings can be connected through a logarithmic sequence of compatible matchings and proving related conjectures for specific cases.

Contribution

It introduces a logarithmic bound on transforming one perfect matching into another via compatible matchings and advances the understanding of edge-disjoint compatible matchings.

Findings

Any two perfect matchings can be connected through O(log n) compatible matchings.

Proved the strongest form of a conjecture for matchings with vertical and horizontal segments.

Established that every perfect matching has a compatible matching with about 80% of its edges.

Abstract

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are \emph{compatible} if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings $M$ and $M'$ of the same set of $n$ points, for some $k\in\Oh{\log n}$, there is a sequence of perfect matchings $M=M_0,M_1,...,M_k=M'$, such that each $M_i$ is compatible with $M_{i+1}$. This improves the previous best bound of $k\leq n-2$. We then study the conjecture: \emph{every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching}. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with $n$ edges has an edge-disjoint compatible matching with approximately $4n/5$ edges.