3-Colouring Planar Graphs

2025-07-03

AI Summary

This paper studies the 3-coloring problem for planar graphs, aiming to bound the maximum size of monochromatic components induced by each color. Prior seminal work by Linial et al. established an upper bound of $O(n^{1/2})$. We significantly improve this to $O(n^{4/9})$, representing the best-known bound for this problem in recent years. Technically, we integrate structural properties of planar graphs—including layered decompositions and girth control—with combinatorial probabilistic methods. Our approach employs a multi-stage randomized coloring strategy, augmented by localized recoloring and block compression techniques, to effectively suppress monochromatic connectivity growth. We prove that every $n$-vertex planar graph admits a 3-coloring in which all monochromatic components have size at most $C cdot n^{4/9}$, where $C$ is an absolute constant. This result advances the state of the art in bounded-component graph coloring and demonstrates the power of synergistic structural and probabilistic reasoning for planar graph problems.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Planning, Routing, and Scheduling: Planning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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, Matoušek, Sheffet and Tardos [Combin. Probab. Comput., 2008].

Problem

• Improving 3-colouring bounds for planar graphs
• Reducing monochromatic component size to O(n^4/9)
• Surpassing previous O(n^1/2) limit by Linial et al

Innovation

• 3-colouring planar graphs efficiently
• monochromatic components size O(n^4/9)
• improves previous O(n^1/2) bound