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

TL;DR

This paper proves the conjecture that 3-cut complexes of squared cycle graphs are shellable, confirming their topological structure and Betti number properties, which was previously conjectured by Bayer et al.

Contribution

The paper establishes the shellability of 3-cut complexes of squared cycle graphs, confirming conjectures about their topological and algebraic properties.

Findings

Proved shellability of 3-cut complexes of squared cycle graphs.

Confirmed conjectures on Betti numbers for these complexes.

Validated topological structure predictions for these 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 $\sigma$ of the vertex set $V(G)$ of $G$ such that the induced subgraph of $G$ on $V(G) \setminus \sigma$ 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 $k=3$.