A lower bound on the acyclic matching number of subcubic graphs

TL;DR

This paper establishes a lower bound of one-sixth of the edges for the acyclic matching number in connected subcubic graphs, with only two small exceptions, advancing understanding of matchings in such graphs.

Contribution

It provides a new lower bound on the acyclic matching number of subcubic graphs, identifying the bound as at least one-sixth of the edges, with specific small exceptions.

Findings

Acyclic matching number ≥ m/6 for most connected subcubic graphs

Two small exceptions where the bound does not hold

Improves bounds on matchings in subcubic graphs

Abstract

The acyclic matching number of a graph $G$ is the largest size of an acyclic matching in $G$, that is, a matching $M$ in $G$ such that the subgraph of $G$ induced by the vertices incident to an edge in $M$ is a forest. We show that the acyclic matching number of a connected subcubic graph $G$ with $m$ edges is at least $m/6$ except for two small exceptions.