Coloring curves on surfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2016
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| _version_ | 1866914707639631872 |
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| author | Gaster, Jonah Greene, Joshua Evan Vlamis, Nicholas G. |
| author_facet | Gaster, Jonah Greene, Joshua Evan Vlamis, Nicholas G. |
| contents | We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1608_01589 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Coloring curves on surfaces Gaster, Jonah Greene, Joshua Evan Vlamis, Nicholas G. Geometric Topology Combinatorics 57M15, 05C15 We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that represent a fixed non-zero homology class is uniquely t-colorable, where t denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry. |
| title | Coloring curves on surfaces |
| topic | Geometric Topology Combinatorics 57M15, 05C15 |
| url | https://arxiv.org/abs/1608.01589 |