Shellability of 3-cut complexes of powers of cycle graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914417510187008 |
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| author | Chauhan, Pratiksha Shukla, Samir |
| author_facet | Chauhan, Pratiksha Shukla, Samir |
| contents | 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 $Δ_k(G)$, 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 $G[V(G) \setminus σ]$ 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 $Δ_3(C_n^p)$ 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 $Δ_3(C_n^p)$. 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 $Δ_3(C_n^p)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23155 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shellability of 3-cut complexes of powers of cycle graphs Chauhan, Pratiksha Shukla, Samir Combinatorics Algebraic Topology 57M15, 52B22, 55U05, 05C69, 05E45 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 $Δ_k(G)$, 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 $G[V(G) \setminus σ]$ 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 $Δ_3(C_n^p)$ 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 $Δ_3(C_n^p)$. 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 $Δ_3(C_n^p)$. |
| title | Shellability of 3-cut complexes of powers of cycle graphs |
| topic | Combinatorics Algebraic Topology 57M15, 52B22, 55U05, 05C69, 05E45 |
| url | https://arxiv.org/abs/2603.23155 |