# Matching points with disks with a common intersection

## TL;DR

This paper proves that for two equal-sized point sets, there exists a perfect matching where the smallest disks through matched pairs all intersect, and this holds for maximum weight matchings as well.

## Contribution

The paper establishes the existence of a perfect matching with intersecting diametral disks for any equal-sized point sets, including maximum weight matchings.

## Key findings

- Existence of a perfect matching with intersecting diametral disks.
- This property holds for maximum weight perfect matchings.
- The result applies to any two equal-sized point sets.

## Abstract

We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p in R and q in B, we consider the disk through p and q with the smallest diameter. We prove that for any R and B such that |R|=|B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. In fact, our result is stronger, and shows that a maximum weight perfect matching has this property.