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COLORING CURVES ON SURFACES

  • JONAH GASTER (a1), JOSHUA EVAN GREENE (a2) and NICHOLAS G. VLAMIS (a3)
Abstract

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 nonzero 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.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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