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 $(\mathrm{\mid }V(G)\mathrm{\mid }-k)$-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 $k\ge 3$, 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 $k=3$.