Simplicial complexes with many facets are vertex decomposable
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916507994292224 |
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| author | Dochtermann, Anton Nair, Ritika Schweig, Jay Van Tuyl, Adam Woodroofe, Russ |
| author_facet | Dochtermann, Anton Nair, Ritika Schweig, Jay Van Tuyl, Adam Woodroofe, Russ |
| contents | Suppose $Δ$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $Δ$ is at least $\binom{n}{c}-2c+1$, then $Δ$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $Δ$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_07316 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplicial complexes with many facets are vertex decomposable Dochtermann, Anton Nair, Ritika Schweig, Jay Van Tuyl, Adam Woodroofe, Russ Combinatorics Commutative Algebra 05E40, 05E45, 13F55 Suppose $Δ$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $Δ$ is at least $\binom{n}{c}-2c+1$, then $Δ$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $Δ$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature. |
| title | Simplicial complexes with many facets are vertex decomposable |
| topic | Combinatorics Commutative Algebra 05E40, 05E45, 13F55 |
| url | https://arxiv.org/abs/2403.07316 |