Shellability of 3-cut complexes of squared cycle graphs

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. (SIAM J Discrete Math 38(2):1630–1675, 2024). In the same article, Bayer et al. conjectured that for \(k \ge 3\), 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 \(k=3\).