Induced matchings in graphs of degree at most 4

TL;DR

This paper proves that connected graphs with maximum degree at most 4, excluding a specific graph, have a strong matching number at least one-ninth of their vertices, and provides a polynomial algorithm to find such matchings.

Contribution

It establishes a tight lower bound on the strong matching number for graphs of degree at most 4 and introduces a polynomial-time algorithm for finding these matchings.

Findings

Bound of 1/9n(G) for strong matching number in degree ≤ 4 graphs

Excludes the specific graph C_{2,5} from the bound

Provides a polynomial-time algorithm to find induced matchings of this size

Abstract

We show that if $G$ is a connected graph of maximum degree at most $4$, which is not $C_{2,5}$, then the strong matching number of $G$ is at least $\frac{1}{9}n(G)$. This bound is tight and the proof implies a polynomial time algorithm to find an induced matching of this size.