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 of the vertex set V (G) of G such that the induced subgraph of G on V (G) 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.