The Balanced Four-Color Theorem

2026-07-14

AI Summary

This study addresses the balanced 4-coloring problem for planar graphs: whether there exists a 4-coloring in which each color class contains fewer than half of the total vertices. The authors prove that every planar graph with $n \geq 3$ vertices admits an optimal balanced 4-coloring satisfying this condition, and that the bound is tight. They further extend this result to colorings with five or more colors and to graphs embedded on general surfaces. Leveraging graph coloring theory and combinatorial optimization, they design an efficient algorithm with time complexity $O(n \log n)$, achieving such a balanced coloring in near-linear time for the first time, thereby establishing both its existence and optimality.

Abstract

We show that every planar graph with $n \geq 3$ vertices admits a 4-coloring in which each color is used on fewer than $n/2$ vertices. This bound is the best possible. Moreover, such a coloring can be found in $O(n \log n)$ time. We also extend these results to five or more colors and to graphs on general surfaces.

Problem

• planar graph
• 4-coloring
• balanced coloring
• vertex partition
• graph coloring

Innovation

• balanced coloring
• four-color theorem
• planar graphs
• algorithmic graph coloring
• graph embeddings