Bichromatic Perfect Matchings with Crossings

TL;DR

This paper investigates the maximum number of crossings in bichromatic perfect matchings for points in convex position, establishing tight bounds and demonstrating the existence of point sets with specific crossing numbers.

Contribution

It provides tight bounds on the maximum crossings in bichromatic perfect matchings for convex point sets, a problem previously unexplored in this form.

Findings

Every convex bichromatic point set admits a perfect matching with at least (3n^2/8 - n/2 + c) crossings.

The established bound is tight, with constructions showing no higher crossing number is always possible.

The crossing number bounds depend on the parameter c, which ranges between -1/2 and 1/8.

Abstract

We consider bichromatic point sets with $n$ red and $n$ blue points and study straight-line bichromatic perfect matchings on them. We show that every such point set in convex position admits a matching with at least $\frac{3n^2}{8}-\frac{n}{2}+c$ crossings, for some $ -\frac{1}{2} \leq c \leq \frac{1}{8}$. This bound is tight since for any $k> \frac{3n^2}{8} -\frac{n}{2}+\frac{1}{8}$ there exist bichromatic point sets that do not admit any perfect matching with $k$ crossings.