Independence Complexes of Hexagonal Grid Graphs

TL;DR

This paper investigates the topological properties of independence complexes in hexagonal grid graphs, revealing their homotopy types for specific cases and providing recursive descriptions for their structure.

Contribution

It determines the homotopy types of independence complexes for hexagonal grid graphs with small widths, extending understanding beyond square grid cases.

Findings

Independence complex of $H_{1 imes 1 imes n}$ is homotopy equivalent to a wedge of two $n$-spheres.

Recursive descriptions for $m=2$ and $m=3$ cases fully determine the homotopy types.

Uses link, deletion operations, and fold lemma for topological analysis.

Abstract

The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively little is known about planar hexagonal grid graphs. In this article, we study the topology of the independence complexes of hexagonal grid graphs $H_{1 \times m \times n}$. For $ m=1, 2, 3$ and $n\geq 1$, we determine their homotopy types. In particular, we show that the independence complex of the hexagonal line tiling $H_{1 \times 1 \times n}$ is homotopy equivalent to a wedge of two $n$-spheres, and for $m=2$ and $m=3$, we obtain recursive descriptions that completely determine the spheres appearing in the homotopy type. Our proofs rely on link and deletion operations, the fold lemma, and a detailed analysis of induced subgraphs.