3-Colouring Planar Graphs

TL;DR

This paper proves that every planar graph with n vertices can be colored with three colors such that each monochromatic component has size proportional to n^{4/9}, improving previous bounds.

Contribution

It establishes a new bound on monochromatic component size in 3-colorings of planar graphs, advancing understanding of graph coloring properties.

Findings

Monochromatic components are of size O(n^{4/9})

Improves previous bound of O(n^{1/2})

Advances theoretical understanding of planar graph colorings

Abstract

We show that every $n$-vertex planar graph is 3-colourable with monochromatic components of size $O(n^{4/9})$. The best previous bound was $O(n^{1/2})$ due to Linial, Matou\v{s}ek, Sheffet and Tardos [Combin. Probab. Comput., 2008].