Factorially many maximum matchings close to the Erd\H{o}s-Gallai bound

TL;DR

This paper investigates the number of maximum matchings in graphs near the Erdős-Gallai bound, showing factorial growth when the graph's size is close to this maximum.

Contribution

It establishes that graphs with size close to the Erdős-Gallai maximum have factorially many maximum matchings, extending understanding of matching abundance.

Findings

Graphs near the Erdős-Gallai bound have factorially many maximum matchings.

Maximum matchings are abundant in graphs with size close to the extremal limit.

The result quantifies the richness of maximum matchings in near-extremal graphs.

Abstract

A classical result of Erd\H{o}s and Gallai determines the maximum size $m(n,\nu)$ of a graph $G$ of order $n$ and matching number $\nu n$. We show that $G$ has factorially many maximum matchings provided that its size is sufficiently close to $m(n,\nu)$.