Shellability of $3$-Cut Complexes of Squared Cycle Graphs

TL;DR

The paper addresses the shellability and homotopy type of the $k$-cut complexes Δ_k(C_n^2) for squared cycle graphs, confirming a conjecture for the case k=3. It develops a tailored shelling order and shows that for $n≥9$, Δ_3(C_n^2) is shellable and homotopy equivalent to a wedge of $\binom{n-4}{2}-9$ spheres of dimension $n-4$, providing explicit Betti-number-like information. The method combines combinatorial shelling with a careful counting of spanning facets to deduce the wedge multiplicity, and connects to broader Fröberg-type and duality results. The work also includes Sage-based explorations of Δ_k(C_n^p) for powered cycles, proposing several conjectures for larger $k$ and $p$ and outlining future directions in the topology of graph cut complexes.

Abstract

For a positive integer $k$, the $k$-cut complex of a graph $G$ is the simplicial complex whose facets are the $(|V(G)|-k)$-subsets $σ$ of the vertex set $V(G)$ of $G$ such that the induced subgraph of $G$ on $V(G) \setminus σ$ is disconnected. These complexes first appeared in the master thesis of Denker and were further studied by Bayer et al.\ in [Topology of cut complexes of graphs, SIAM Journal on Discrete Mathematics, 2024]. In the same article, Bayer et al.\ conjectured that for $k \geq 3$, the $k$-cut complexes of squared cycle graphs are shellable. Moreover, they also conjectured about the Betti numbers of these complexes when $k=3$. In this article, we prove these conjectures for $k=3$.

Theorems & Definitions (46)

• Theorem 1.1: Froberg1990, Eagon1998
• Theorem 1.2: Eagon1998
• Conjecture 1.3: Bayer2024Cutcomplex
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 3.1