The topology of independence complexes of square grids

TL;DR

This paper proves that the independence complexes of m-by-n square grid graphs with certain boundary conditions are homotopy equivalent to wedges of spheres, advancing understanding of their topological structure and confirming a conjecture.

Contribution

It establishes the homotopy type of independence complexes for grid graphs with open or cylindrical boundaries, confirming a conjecture and enriching topological graph theory.

Findings

Independence complexes are homotopy equivalent to wedges of spheres.

Results apply to grids with open or cylindrical boundary conditions.

Confirms a conjecture by Iriye.

Abstract

The independence complex of a graph G is a simplicial complex whose simplices are the independent sets in G. In the last couple of decades, the independence complexes of square grids (with various boundary conditions) have gained much attention because of their connections with the hard square model from statistical physics. In this article, we prove that if G is an $m\times n$ grid with open or cylindrical boundary condition then its independence complex is homotopy equivalent to a wedge of spheres. A part of this result settles a conjecture of Iriye.