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Independence Complexes of Hexagonal Grid Graphs

Himanshu Chandrakar, Anurag Singh

TL;DR

This work investigates independence complexes of planar hexagonal grid graphs $H_{1\times m\times n}$ for $m\in\{1,2,3\}$ and $n\ge1$, revealing distinct homotopy behaviours across the cases. Using a toolkit of link/deletion, fold lemmas, and induced-subgraph decompositions, the authors derive explicit homotopy types: for $m=1$, $\operatorname{Ind}(H_{1\times1\times n}) \simeq \mathbb{S}^n \vee \mathbb{S}^n$, while for $m=2,3$ they provide recursive descriptions that determine the spheres appearing in the homotopy type. Central constructions include the induced subgraphs $Y_n$ and $H_n^2$, with detailed recurrences $\operatorname{Ind}(Y_n)$ and $\operatorname{Ind}(H_n^2)$ expressed via suspensions $\Sigma^k$ of smaller complexes. The results advance understanding of independence complexes on hexagonal grids and demonstrate how geometric structure influences topological type.

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.

Independence Complexes of Hexagonal Grid Graphs

TL;DR

This work investigates independence complexes of planar hexagonal grid graphs for and , revealing distinct homotopy behaviours across the cases. Using a toolkit of link/deletion, fold lemmas, and induced-subgraph decompositions, the authors derive explicit homotopy types: for , , while for they provide recursive descriptions that determine the spheres appearing in the homotopy type. Central constructions include the induced subgraphs and , with detailed recurrences and expressed via suspensions of smaller complexes. The results advance understanding of independence complexes on hexagonal grids and demonstrate how geometric structure influences topological type.

Abstract

The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of . 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 . For and , we determine their homotopy types. In particular, we show that the independence complex of the hexagonal line tiling is homotopy equivalent to a wedge of two -spheres, and for and , 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.
Paper Structure (9 sections, 17 theorems, 63 equations, 30 figures)

This paper contains 9 sections, 17 theorems, 63 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics of graph theory
  4. Labelling of vertices in the general hexagonal grid graph.
  5. Topological preliminaries and tools for independence complexes
  6. Independence Complex of $H_{1\times 1 \times n}$
  7. Independence Complex of $H_{1\times 2 \times n}$
  8. Independence complex of $Y_n$
  9. Independence complex of $H_{n}^{2}$

Key Result

Theorem 1.1

For $n \geq 1$, the independence complex of the hexagonal line tiling $H_{1 \times 1 \times n}$ is homotopy equivalent to a wedge of two $n$-dimensional spheres, that is,

Figures (30)

  • Figure 1: The graph $H_{1 \times 4 \times 6}$.
  • Figure 2: The graph $H_{1 \times 1 \times n}$.
  • Figure 3: The graph $X_n^{(1)}$
  • Figure 4: The graph $X_{n}^{(2)}$.
  • Figure 5: Foldings in the graph $X_n^{(1)}$
  • ...and 25 more figures

Theorems & Definitions (27)

  • Theorem 1.1: \ref{['thm: homotopy type of H11n']}
  • Definition 2.1: kozlov2008combinatorial
  • Lemma 2.1: ADAMASZEK20121031
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4: Barmakstarclusters
  • Theorem 2.5: KozlovDirectedTrees
  • Theorem 2.6: KozlovDirectedTrees
  • Theorem 3.1
  • Lemma 3.2
  • ...and 17 more