Coloring curves on surfaces

TL;DR

This paper investigates the chromatic number of the curve graph on surfaces, establishing growth rates, coloring properties, and connections to other mathematical structures, thereby providing improved bounds and insights into surface topology.

Contribution

It introduces new bounds on the chromatic number of the curve graph and explores coloring properties for specific classes of curves, connecting these to Kneser graphs and hyperbolic geometry.

Findings

Chromatic number of separating curve graph grows like k log k

Graph of curves in a fixed homology class is uniquely t-colorable

Exact results obtained for low complexity surfaces

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 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.