Uniquely restricted matchings in subcubic graphs

TL;DR

This paper investigates the size of uniquely restricted matchings in subcubic graphs, proposing bounds based on graph parameters and confirming parts of a conjecture related to girth and matching size.

Contribution

It introduces new bounds for uniquely restricted matchings in subcubic graphs, including a conjecture and partial proof for graphs with large girth.

Findings

Established a lower bound of (m+b)/6 for certain subcubic graphs

Proved that graphs with girth at least 7 have a matching of size at least (n-1)/3

Partially confirmed a conjecture by Fürst and Rautenbach

Abstract

A matching $M$ in a graph $G$ is uniquely restricted if no other matching in $G$ covers the same set of vertices. We conjecture that every connected subcubic graph with $m$ edges and $b$ bridges that is distinct from $K_{3,3}$ has a uniquely restricted matching of size at least $\frac{m+b}{6}$, and we establish this bound with $b$ replaced by the number of bridges that lie on a path between two vertices of degree at most $2$. Moreover, we prove that every connected subcubic graph of order $n$ and girth at least $7$ has a uniquely restricted matching of size at least $\frac{n-1}{3}$, which partially confirms a Conjecture of F\"{u}rst and Rautenbach (Some bounds on the uniquely restricted matching number, arXiv:1803.11032).