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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.16838 |
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| author | Pineda-Villavicencio, Guillermo Wang, Jie Yost, David |
| author_facet | Pineda-Villavicencio, Guillermo Wang, Jie Yost, David |
| contents | We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are not realised by any polytope. For $d$-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polytopes with low excess degree Pineda-Villavicencio, Guillermo Wang, Jie Yost, David Combinatorics 52B11 We study the existence and structure of $d$-polytopes for which the number $f_1$ of edges is small compared to the number $f_0$ of vertices. Our results are more elegantly expressed in terms of the excess degree of the polytope, defined as $2f_1-df_0$. We show that the excess degree of a $d$-polytope cannot lie in the range $[d+3,2d-7]$, complementing the known result that values in the range $[1,d-3]$ are impossible. In particular, many pairs $(f_0,f_1)$ are not realised by any polytope. For $d$-polytopes with excess degree $d-2$, strong structural results are known; we establish comparable results for excess degrees $d$, $d+2$, and $2d-6$. Frequently, in polytopes with low excess degree, say at most $2d-6$, the nonsimple vertices all have the same degree and they form either a face or a missing face. We show that excess degree $d+1$ is possible only for $d=3,5$, or $7$, complementing the known result that an excess degree $d-1$ is possible only for $d=3$ or $5$. |
| title | Polytopes with low excess degree |
| topic | Combinatorics 52B11 |
| url | https://arxiv.org/abs/2405.16838 |