The average size of maximal matchings in graphs

TL;DR

This paper studies the ratio of the average size of maximal matchings to maximum matchings in graphs, introducing a technique to analyze its asymptotic behavior across various graph classes.

Contribution

It presents a new general technique for determining the asymptotic behavior of the average maximal matching size ratio in different graph families.

Findings

The ratio approaches 1 when many maximal matchings are near maximum size.

The ratio approaches 0.5 when many maximal matchings are small.

The technique recovers known results and finds new asymptotic values for various graph classes.

Abstract

We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1. Conversely, if many maximal matchings have a small size, $\avM(G)$ approaches $\frac{1}{2}$. We propose a general technique to determine the asymptotic behavior of $\avM(G)$ for various classes of graphs. To illustrate the use of this technique, we first show how it makes it possible to find known asymptotic values of $\avM(G)$ which were typically obtained using generating functions, and we then determine the asymptotic value of $\avM(G)$ for other families of graphs, highlighting the spectrum of possible values of this graph invariant between $\frac{1}{2}$ and $1$.