$3$-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 σ of the vertex set $V(G)$ of G such that the induced subgraph of G on V(G) 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 ≥ 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 $k=3$.