Topology of total cut complexes and cut complexes of grid graphs

TL;DR

The paper addresses the topology of total $k$-cut complexes $\Delta_k^t(G)$ and cut complexes $\Delta_k(G)$ on grid graphs, deriving explicit homotopy descriptions and shellability results. It employs discrete Morse theory and a framework of link/deletion analysis to inductively determine the homotopy types for the $2\times n$ and $3\times n$ grid families, and proves shellability for the $2\times n$ case across relevant $k$-ranges. Specifically, it shows $\Delta_{k,n}^t \simeq \bigvee_{\binom{n-1}{k-1}} \mathbb{S}^{2n-2k}$ for $2 \le k \le n$ on $2\times n$ grids, and $\Delta_n^t \simeq \bigvee_{\binom{2n-2}{2}} \mathbb{S}^{3n-6}$ on $3\times n$ grids, while establishing shellability of $\Delta_k(\mathcal{G}_{2\times n})$ for $3 \le k \le 2n-2$ and $n \ge 3$. These results confirm conjectures in the grid-graph setting and advance understanding of how these graph complexes behave topologically in structured graph families.

Abstract

Inspired by the work of Fr{ö}berg (1990) and Eagon and Reiner (1998), Bayer et al. recently introduced two new graph complexes: total cut complexes and cut complexes. In this article, we investigate these complexes specifically for (rectangular) grid graphs, focusing on $2 \times n$ and $3 \times n$ cases. We extend and refine the work of Bayer et al., proving and strengthening several of their conjectures, thereby enhancing the understanding of these graph complexes' topological and combinatorial properties.

Figures (12)

• Figure 1: The graph $\mathcal{G}_{2\times n}$.
• Figure 2: The graph $\mathcal{G}_{2\times n}'$.
• Figure 3: Total $3-$cut complex of the graph $G_1$.
• Figure 4: $3-$cut complex of the graph $G_2$.
• Figure 5: The graph $\mathcal{G}_{3\times n}$.

Theorems & Definitions (60)

• Conjecture 1.1: bayer2024total, Conjecture 5.2
• Theorem 1.1: Theorems \ref{['thm: homotopy type of total k-cut complex of 2n-gg']} and \ref{['thm: homotopy of 3-cut complex of (3×n)-gg']}
• Conjecture 1.2: bayer2024cutB, Conjecture 5.9
• Theorem 1.2: \ref{["lemma: Shellability of 2n'gg"]}
• Definition 2.1
• Definition 2.2
• Lemma 2.1: Matsushitantimes4, Lemma 2.1
• Lemma 2.2: BjornerTopoMethods, Lemma 10.4(b)
• Definition 2.3: kozlovCombAlgTopo, Defintion 12.1
• Definition 2.4: jonsson2008simplicial, Definition 3.25