Total $2$-cut complexes of powers of cycle graphs and Cartesian products of certain graphs

TL;DR

This paper analyzes total 2-cut complexes Δ_2^t(G) for three graph families: p-th powers of cycle graphs C_n^p and Cartesian products with K_m and either P_n or C_n. Employing discrete Morse theory, the authors show these complexes are homotopy equivalent to wedges of spheres and determine the sphere dimensions and multiplicities, resolving Shen et al.'s conjecture for Δ_2^t(C_n^p). They provide explicit wedge counts: Δ_2^t(K_m□P_n) ≃ ∨^{(m-1)(n-1)} S^{mn-4} and Δ_2^t(K_m□C_n) ≃ ∨^{n(m-1)+1} S^{mn-4}. The work highlights a general technique for turning combinatorial graph data into topological invariants and sets the stage for exploring higher-cut variants and broader graph classes, with supporting computational data guiding future conjectures.

Abstract

For a positive integer $k$, the \emph{ total $k$-cut complex} of a graph $G$, denoted as $Δ_k^t(G)$, is the simplicial complex whose facets are $σ\subseteq V(G)$ such that $|σ| = |V(G)|-k$ and the induced subgraph $G[V(G) \setminus σ]$ does not contain any edge. These complexes were introduced by Bayer et al.\ in \cite{Bayer2024TotalCutcomplex} in connection with commutative algebra. In the same paper, they studied the homotopy types of these complexes for various families of graphs, including cycle graphs $C_n$, squared cycle graphs $C_n^2$, and Cartesian products of complete graphs and path graphs $K_m \square P_2$ and $K_2 \square P_n$. In this article, we extend the work of Bayer et al.\ for these families of graphs. We focus on the complexes $Δ_2^t(G)$ and determine the homotopy types of these complexes for three classes of graphs: (i) $p$-th powers of cycle graphs $C_n^p$ (ii) $K_m \square P_n$ and (iii) $K_m \square C_n$. Using discrete Morse theory, we show that these complexes are homotopy equivalent to wedges of spheres. We also give the number and dimension of spheres appearing in the homotopy type. Our result on powers of cycle graphs $C_n^p$ proves a conjecture of Shen et al.\ about the homotopy type of the complexes $Δ_2^t(C_n^p)$.

Figures (1)

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Theorems & Definitions (44)

• Example 1.1
• Conjecture 1.2: shen2025homotopy
• Theorem 1.3: Theorem \ref{['theorem:powers_of_cycle']}
• Theorem 1.4: Theorem \ref{['theorem:K_m[]P_n']}
• Theorem 1.5: Theorem \ref{['theorem:K_m[]C_n']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5: JonssonBook