Induced Matchings in Graphs of Bounded Maximum Degree

TL;DR

This paper establishes a tight lower bound on the induced matching number for graphs with large maximum degree and no isolated vertices, and provides a polynomial-time algorithm to find such matchings.

Contribution

The paper proves a new sharp lower bound on induced matchings in graphs of bounded maximum degree and introduces an efficient algorithm to find these matchings.

Findings

Established a tight lower bound on induced matchings for large-degree graphs.

Provided a polynomial-time algorithm to compute induced matchings matching the bound.

Validated the bound's sharpness through theoretical proof.

Abstract

For a graph $G$, let $\nu_s(G)$ be the induced matching number of $G$. We prove that $\nu_s(G) \geq \frac{n(G)}{(\lceil\frac{\Delta}{2}\rceil+1) (\lfloor\frac{\Delta}{2}\rfloor+1)}$ for every graph of sufficiently large maximum degree $\Delta$ and without isolated vertices. This bound is sharp. Moreover, there is polynomial-time algorithm which computes induced matchings of size as stated above.