# Finding big matchings in planar graphs quickly

## TL;DR

This paper presents a linear-time algorithm for finding large matchings in planar graphs with minimum degree 3, improving upon previous methods that relied on general maximum matching algorithms with higher complexity.

## Contribution

It introduces a novel linear-time algorithm that efficiently finds large matchings in planar graphs, bypassing the need for complex maximum matching computations.

## Key findings

- Achieves linear-time complexity for the problem
- Guarantees a matching of size at least n/3 in such graphs
- Improves practical efficiency over previous algorithms

## Abstract

It is well-known that every $n$-vertex planar graph with minimum degree 3 has a matching of size at least $\frac{n}{3}$. But all proofs of this use the Tutte-Berge-formula for the size of a maximum matching. Hence these proofs are not directly algorithmic, and to find such a matching one must apply a general-purposes maximum matching algorithm, which has run-time $O(n^{1.5}\alpha(n))$ for planar graphs. In contrast to this, this paper gives a linear-time algorithm that finds a matching of size at least $\frac{n}{3}$ in any planar graph with minimum degree 3.