Independence Complexes of Hexagonal Grid Graphs

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.