Uniquely restricted matchings in subcubic graphs without short cycles

TL;DR

This paper proves that most connected subcubic graphs with girth at least 5 have a large uniquely restricted matching, except for two specific cubic graphs, advancing understanding of matchings in such graphs.

Contribution

It establishes a lower bound on the size of uniquely restricted matchings in connected subcubic graphs with girth at least 5, identifying two exceptions.

Findings

Most such graphs have a uniquely restricted matching of size at least (n-1)/3.

Two specific cubic graphs of orders 14 and 20 are exceptions.

The result improves bounds on matchings in restricted graph classes.

Abstract

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We prove that any connected subcubic graph with $n$ vertices and girth at least $5$ contains a uniquely restricted matching of size at least $(n-1) / 3$ except for two exceptional cubic graphs of order $14$ and $20$.