Some bounds on the maximum induced matching numbers of certain grids

TL;DR

This paper establishes upper bounds on the maximum induced matching numbers for certain grid graphs with specific size constraints, advancing understanding of their combinatorial properties.

Contribution

It provides new upper bounds for the maximum induced matching numbers in grids with particular dimensions and modular conditions, which was previously unexplored.

Findings

Derived upper bounds for grid graphs with n,m ≥ 9

Focused on grids where m ≡ 1 mod 4 and nm is odd

Enhanced theoretical understanding of induced matchings in grids

Abstract

An induced matching $M$ in a graph $G$ is a matching in $G$ that is also the edge set of an induced subgraph of $G$. That is, any edge not in $M$ must have no more than one incident vertex saturated by $M$. The maximum size $|M|$ of an induced matching $M$ of $G$ is maximum induced matching number of $G$, which is denoted by $\textrm{Max}(G)$. In this article, we obtain upper bounds for $\textrm{Max}(G)$, for $G=G_{n,m}$, grids with $n,m \geq 9$, $m\equiv 1 \mod 4$ and $nm$ odd.