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arXiv · 2406.01979

Shellability of $3$-Cut Complexes of Squared Cycle Graphs

Pratiksha Chauhan·Samir Shukla·Kumar Vinayak
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Abstract

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 $\sigma$ 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 $k \geq 3$, 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 $k=3$.

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Pratiksha Chauhan, Samir Shukla, Kumar Vinayak. 2024-06-04. Shellability of $3$-Cut Complexes of Squared Cycle Graphs. https://arxiv.org/abs/2406.01979

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