On Compatible Matchings

TL;DR

This paper investigates the maximum size of crossing-free matchings compatible with multiple labeled point sets in the plane, providing bounds and constructions for various numbers of sets.

Contribution

It introduces new bounds on the size of compatible matchings for multiple point sets, including both lower and upper bounds, and explores the minimum number of copies needed for trivial matchings.

Findings

Existence of compatible matchings with (n) edges for two convex sets.

Construction of (n^{1/}) size matchings for  labeled point sets.

Probabilistic upper bounds showing matchings of size (n^{2/(+1)}) for any labeling.

Abstract

A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $\lfloor \sqrt {2n}\rfloor$ edges. More generally, for any $\ell$ labeled point sets we construct compatible matchings of size $\Omega(n^{1/\ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $\ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${\mathcal{O}}(n^{2/({\ell}+1)})$ edges. Finally, we show that $\Theta(\log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.