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COLORING CURVES ON SURFACES

Part of: Low-dimensional topology Graph theory

Published online by Cambridge University Press:  04 September 2018

JONAH GASTER ,
JOSHUA EVAN GREENE  and
NICHOLAS G. VLAMIS
JONAH GASTER
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada; jbgaster@gmail.com
JOSHUA EVAN GREENE
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA; joshua.greene@bc.edu
NICHOLAS G. VLAMIS
Affiliation:
Department of Mathematics, Queens College of CUNY, Flushing, NY 11367, USA; nicholas.vlamis@qc.cuny.edu

Abstract

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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.

MSC classification

Primary: 57M15: Relations with graph theory
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e17
Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
© The Author(s) 2018

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