Matching complexes of small grids

TL;DR

This paper investigates the topological structure of matching complexes of small grid graphs, showing they are wedges of spheres, thus advancing understanding of their combinatorial and topological properties.

Contribution

The paper proves that the independence complex of a family of generalized line graphs of small grids is a wedge of spheres, addressing a problem posed by Braun and Hough.

Findings

Independence complex of Δ^m_n is a wedge of spheres

Provides a topological characterization of matching complexes of small grids

Addresses a previously open problem in the field

Abstract

The matching complex $M(G)$ of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. The matching complex $M(G)$ is isomorphic to the independence complex of the line graph $L(G)$. Braun and Hough introduced a family of graphs $\Delta^m_n$, which is a generalization of the line graph of the $(n \times 2)$-grid graph. In this paper, we show that the independence complex of $\Delta^m_n$ is a wedge of spheres. This gives an answer to a problem suggested by Braun and Hough.