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On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs

Himanshu Chandrakar

TL;DR

This work addresses the topology of 3-cut complexes of hexagonal grid graphs by constructing an explicit shelling order, showing $\Delta_3\left(H_{1\times m\times n}\right)$ is shellable for all $m,n\ge1$. Using reverse lexicographic facets and a carefully chosen family of exceptional facets moved to the end, the authors obtain a concrete shelling and a precise count of spanning facets, enabling a direct identification of the homotopy type as a wedge of $\psi_{m,n}$ spheres of dimension $d=2m+2n+2mn-4$. The key contribution is the combinatorial shelling construction that yields the exact formula $\psi_{m,n}=\binom{2m+2n+2mn-1}{2}-\big[(6m+2)n+(2m-4)\big]$, providing an explicit, constructive path from shellability to topological decomposition. This work not only confirms known general results but also offers a hands-on counting method with potential generalizations to higher $k$ and broader hexagonal tilings, impacting both combinatorial topology and graph-theoretic insights for grid-based complexes.

Abstract

The $k$-cut complex was recently introduced by Bayer et al. as a generalization of earlier work of Fr{ö}berg (1990) and Eagon and Reiner (1998), and was shown to be shellable for several classes of graphs. In this article, we prove that the $3$-cut complexes of the hexagonal grid graphs $H_{1 \times m \times n}$ are shellable for all $m,n \geq 1$, by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by $ψ_{m,n}$, and deduce that the complex is homotopy equivalent to a wedge of $ψ_{m,n}$ spheres of dimension $\left( 2m + 2n + 2mn - 4 \right)$, where $$ψ_{m,n} = \binom{2m+2n+2mn-1}{2} - \left[ \left( 6m+2 \right) n + (2m-4) \right].$$ While these topological properties can be obtained from general results of Bayer et al., we provide an explicit combinatorial construction of a shelling order, yielding a direct counting formula for the number of spheres in the wedge sum decomposition.

On Shellability of 3-Cut Complexes of Hexagonal Grid Graphs

TL;DR

This work addresses the topology of 3-cut complexes of hexagonal grid graphs by constructing an explicit shelling order, showing is shellable for all . Using reverse lexicographic facets and a carefully chosen family of exceptional facets moved to the end, the authors obtain a concrete shelling and a precise count of spanning facets, enabling a direct identification of the homotopy type as a wedge of spheres of dimension . The key contribution is the combinatorial shelling construction that yields the exact formula , providing an explicit, constructive path from shellability to topological decomposition. This work not only confirms known general results but also offers a hands-on counting method with potential generalizations to higher and broader hexagonal tilings, impacting both combinatorial topology and graph-theoretic insights for grid-based complexes.

Abstract

The -cut complex was recently introduced by Bayer et al. as a generalization of earlier work of Fr{ö}berg (1990) and Eagon and Reiner (1998), and was shown to be shellable for several classes of graphs. In this article, we prove that the -cut complexes of the hexagonal grid graphs are shellable for all , by constructing an explicit shelling order using reverse lexicographic ordering. From this shelling, we determine the number of spanning facets, denoted by , and deduce that the complex is homotopy equivalent to a wedge of spheres of dimension , where While these topological properties can be obtained from general results of Bayer et al., we provide an explicit combinatorial construction of a shelling order, yielding a direct counting formula for the number of spheres in the wedge sum decomposition.
Paper Structure (12 sections, 10 theorems, 51 equations, 1 figure)

This paper contains 12 sections, 10 theorems, 51 equations, 1 figure.

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
  6. Shellable simplicial complexes and spanning facets
  7. Shellability of 3-cut complex of hexagonal grid graph
  8. Number of facets in $\Delta_3\left(H_{1\times m\times n}\right)$
  9. Shelling order for the facets of $\Delta_3\left(H_{1\times m\times n}\right)$
  10. Shellability of $\Delta_{3}\left( H_{1\times m \times n}\right)$
  11. Spanning facets and the homotopy type of the 3-cut complex of hexagonal tiling
  12. Future directions

Key Result

Theorem 1.1

For $m, n\geq 1$, the simplicial complex $\Delta_3\left(H_{1\times m \times n}\right)$ is shellable.

Figures (1)

  • Figure 1: The graph $H_{1 \times 4 \times 6}$.

Theorems & Definitions (24)

  • Theorem 1.1: \ref{['thm: shellability of hexagonal grid graphs']}
  • Theorem 1.2: \ref{['thm: spanning facets and homotopy type']}
  • Definition 2.1: Hexagonal grid graphs
  • Definition 2.2: bayer2024cut, Definition 2.7
  • Definition 2.3: kozlov2008combinatorial, Defintion 12.1
  • Definition 2.4: Boundary of a simplex
  • Theorem 2.1: kozlov2008combinatorial
  • Lemma 3.1
  • proof
  • Definition 3.1
  • ...and 14 more