Shellability of 3-cut complexes of powers of cycle graphs

In connection with commutative algebra, Bayer et al. introduced cut complexes in [Topology of cut complexes of graphs, SIAM J.\ Discrete Math., 38(2):1630-1675, 2024]. For a positive integer $k$, the $k$-cut complex of a graph $G$, denoted as $\Delta_k(G)$, 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 $G[V(G) \setminus \sigma]$ is disconnected. Let $C_n^p$ denote the $p$-th power graph of the cycle graph $C_n$ on $n$ vertices. In this article, we show that $\Delta_3(C_n^p)$ is shellable for $n \geq 6p-3$, and therefore these complexes are homotopy equivalent to a wedge of spheres of dimension $n-4$. We provide an explicit shelling order on the facets of $\Delta_3(C_n^p)$. We also characterize and count the number of spanning facets in this shelling order, and determine the number of spheres appearing in the wedge in the homotopy type of $\Delta_3(C_n^p)$.