# Structure of the largest subgraphs of $G_{n,p}$ with a given matching number

## TL;DR

This paper investigates the structure of the largest subgraphs in Erdős-Rényi random graphs with a fixed matching number, extending classical results and identifying regimes where these structures hold with high probability.

## Contribution

It extends Erdős and Gallai's classification of largest subgraphs with a given matching number from complete graphs to Erdős-Rényi graphs under certain probability conditions.

## Key findings

- Classical structure results extend to $G_{n,p}$ when $p o 0$ or $p$ is sufficiently large.
- The structure does not extend with high probability in the intermediate probability range.
- The results specify probability thresholds for the structural extension.

## Abstract

This paper examines the structure of the largest subgraphs of the Erd\H{o}s-R\'enyi random graph, $G_{n,p}$, with a given matching number. This extends a result of Erd\H{o}s and Gallai who, in 1959, gave a classification of the structures of the largest subgraphs of $K_n$ with a given matching number. We show that their result extends to $G_{n,p}$ with high probability when $p\ge \frac{8 \ln n}{n}$ or $p \ll \frac{1}{n}$, but that it does not extend (again with high probability) when $\frac{4\ln(2e)}{n} < p< \frac{\ln n}{3n}$.