On the faces of unigraphic $3$-polytopes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917794505818112 |
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| author | Maffucci, Riccardo W. |
| author_facet | Maffucci, Riccardo W. |
| contents | A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a difficult problem.
In this paper we prove that, apart from pyramids, all unigraphic $3$-polytopes have no $n$-gonal faces for $n\geq 10$. Our method involves defining several planar graph transformations on a given $3$-polytope containing an $n$-gonal face with $n\geq 10$. The delicate part is to prove that, for every such $3$-polytope, at least one of these transformations both preserves $3$-connectivity, and is not an isomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_20012 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the faces of unigraphic $3$-polytopes Maffucci, Riccardo W. Combinatorics 05C10, 05C40, 05C75, 05C76, 05C07, 05C62, 05C85, 52B05, 52B10 A $3$-polytope is a $3$-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other $3$-polytope, up to graph isomorphism. The classification of unigraphic $3$-polytopes appears to be a difficult problem. In this paper we prove that, apart from pyramids, all unigraphic $3$-polytopes have no $n$-gonal faces for $n\geq 10$. Our method involves defining several planar graph transformations on a given $3$-polytope containing an $n$-gonal face with $n\geq 10$. The delicate part is to prove that, for every such $3$-polytope, at least one of these transformations both preserves $3$-connectivity, and is not an isomorphism. |
| title | On the faces of unigraphic $3$-polytopes |
| topic | Combinatorics 05C10, 05C40, 05C75, 05C76, 05C07, 05C62, 05C85, 52B05, 52B10 |
| url | https://arxiv.org/abs/2305.20012 |