Minimal pentagulations of $n$-gons
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915061119844352 |
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| author | Kabenyuk, Mikhail |
| author_facet | Kabenyuk, Mikhail |
| contents | A planar graph $G$ is called a pentagulation of an $n$-gon ($n\geq$ is an integer) if all faces of $G$ are pentagons, except one, which is an $n$-gon. A $3$-connected pentagulation $G$ of an $n$-gon is called minimal if it has the smallest number of pentagons among all such $3$-connected pentagulations. It is known that minimal pentagulations of the $3$-gon and $4$-gon contain 15 and 14 pentagons, respectively. We determined all minimal pentagulations of $n$-gons for all $n$ such that $3\leq n\leq 12$ using computer calculations. The calculations employed the plantri package, which generates all planar triangulations for a given number of vertices. We also present several open questions on this topic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal pentagulations of $n$-gons Kabenyuk, Mikhail Combinatorics 05C10, 05C38 A planar graph $G$ is called a pentagulation of an $n$-gon ($n\geq$ is an integer) if all faces of $G$ are pentagons, except one, which is an $n$-gon. A $3$-connected pentagulation $G$ of an $n$-gon is called minimal if it has the smallest number of pentagons among all such $3$-connected pentagulations. It is known that minimal pentagulations of the $3$-gon and $4$-gon contain 15 and 14 pentagons, respectively. We determined all minimal pentagulations of $n$-gons for all $n$ such that $3\leq n\leq 12$ using computer calculations. The calculations employed the plantri package, which generates all planar triangulations for a given number of vertices. We also present several open questions on this topic. |
| title | Minimal pentagulations of $n$-gons |
| topic | Combinatorics 05C10, 05C38 |
| url | https://arxiv.org/abs/2412.09161 |