COLORING CURVES ON SURFACES. (4th September 2018)

• Record Type: Journal Article
• Authors: GASTER, JONAH; GREENE, JOSHUA EVAN; VLAMIS, NICHOLAS G.
• Is Part Of: Forum of mathematics. Volume 6(2018)
• Publication Date: 2018-09-04
• Subjects: 57M15 (primary), -- 05C15 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2018.12
• Languages: English
• ISSNs: 2050-5094

Abstract

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.