Matching complexes of $\bf 3 \times n$ grid graphs

TL;DR

This paper proves that the matching complexes of 3 by n grid graphs are homotopy equivalent to wedges of spheres and provides a detailed list of the sphere dimensions, extending previous results from 2 by n grids.

Contribution

It establishes the homotopy equivalence of matching complexes of 3 by n grid graphs to wedges of spheres and details the sphere dimensions involved.

Findings

Matching complexes of 3 by n grid graphs are homotopy equivalent to wedges of spheres.

The dimensions of the spheres in the wedge are comprehensively listed.

Extends previous results from 2 by n grid graphs to 3 by n grids.

Abstract

The matching complex of a graph $G$ is a simplicial complex whose simplices are matchings in $G$. In the last few years the matching complexes of grid graphs have gained much attention among the topological combinatorists. In 2017, Braun and Hough obtained homological results related to the matching complexes of $2 \times n$ grid graphs. Further in 2019, Matsushita showed that the matching complexes of $2 \times n$ grid graphs are homotopy equivalent to a wedge of spheres. In this article we prove that the matching complexes of $3\times n$ grid graphs are homotopy equivalent to a wedge of spheres. We also give the comprehensive list of the dimensions of spheres appearing in the wedge.