Chromatic numbers of hyperbolic surfaces

TL;DR

This paper investigates the chromatic numbers of hyperbolic surfaces, establishing bounds that depend on distance and genus, and provides examples to demonstrate the bounds' significance.

Contribution

It introduces new bounds on the $d$-chromatic number of hyperbolic surfaces based on distance and genus, with constructions showing their relevance.

Findings

Upper bounds on $d$-chromatic number depending on $d$

Upper bounds on chromatic number depending on genus $g$

Constructed examples validating the bounds

Abstract

This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the $d$-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance $d$ are of a different color. We prove upper bounds on the $d$-chromatic number of any hyperbolic surface which only depend on $d$. In another direction, we investigate chromatic numbers of closed genus $g$ surfaces and find upper bounds that only depend on $g$ (and not on $d$). For both problems, we construct families of examples that show that our bounds are meaningful.