Chromatic numbers, Buchstaber numbers and chordality of Bier spheres
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917274967867392 |
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| author | Limonchenko, Ivan Vavpetič, Aleš |
| author_facet | Limonchenko, Ivan Vavpetič, Aleš |
| contents | We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d\geq 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere $\mathrm{Bier}(K)$, for each prime $p\in\mathbb{N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_20861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chromatic numbers, Buchstaber numbers and chordality of Bier spheres Limonchenko, Ivan Vavpetič, Aleš Combinatorics Algebraic Topology 05C15, 05E45, 13F55, 52B12, 57S12 We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d\geq 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere $\mathrm{Bier}(K)$, for each prime $p\in\mathbb{N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes. |
| title | Chromatic numbers, Buchstaber numbers and chordality of Bier spheres |
| topic | Combinatorics Algebraic Topology 05C15, 05E45, 13F55, 52B12, 57S12 |
| url | https://arxiv.org/abs/2412.20861 |