Coloring planar graphs with three colors and no large monochromatic components

TL;DR

This paper proves that for any planar graph with maximum degree Δ, there exists a 3-coloring where each monochromatic component is bounded in size by a function of Δ, answering a long-standing open question.

Contribution

It establishes the existence of a function bounding monochromatic component size in 3-colorings of planar graphs with maximum degree Δ, extending to bounded genus graphs, and confirms the optimality of these bounds.

Findings

Monochromatic components are bounded by a function of Δ.

The result applies to graphs of bounded genus.

The bounds are proven to be optimal.

Abstract

We prove the existence of a function $f :\mathbb{N} \to \mathbb{N}$ such that the vertices of every planar graph with maximum degree $\Delta$ can be 3-colored in such a way that each monochromatic component has at most $f(\Delta)$ vertices. This is best possible (the number of colors cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan, and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.