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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2208.05254 |
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| _version_ | 1866909985061994496 |
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| author | Schwartz, Richard Evan |
| author_facet | Schwartz, Richard Evan |
| contents | We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we construct, which are roughly extremal in some sense, are based on a novel geometric interpretation of continued fractions. We also present a conjectural sharp "isoperimetric inequality" for colorings of this kind of triangulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_05254 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Continued Fractions and the 4-Color Theorem Schwartz, Richard Evan Combinatorics Number Theory We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we construct, which are roughly extremal in some sense, are based on a novel geometric interpretation of continued fractions. We also present a conjectural sharp "isoperimetric inequality" for colorings of this kind of triangulation. |
| title | Continued Fractions and the 4-Color Theorem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2208.05254 |