Shellability of $3$-Cut Complexes of Squared Cycle Graphs
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912198742245376 |
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| author | Chauhan, Pratiksha Shukla, Samir Vinayak, Kumar |
| author_facet | Chauhan, Pratiksha Shukla, Samir Vinayak, Kumar |
| contents | 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 $V(G) \setminus σ$ 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 $k \geq 3$, 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 $k=3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01979 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Shellability of $3$-Cut Complexes of Squared Cycle Graphs Chauhan, Pratiksha Shukla, Samir Vinayak, Kumar Combinatorics 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 $V(G) \setminus σ$ 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 $k \geq 3$, 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 $k=3$. |
| title | Shellability of $3$-Cut Complexes of Squared Cycle Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.01979 |