The Balanced Four-Color Theorem

Ken-ichi Kawarabayashi; Hirotaka Yoneda; Masataka Yoneda

The Balanced Four-Color Theorem

Data Structures and Algorithms 2026-07-14 v1 Combinatorics

Authors: Ken-ichi Kawarabayashi , Hirotaka Yoneda , Masataka Yoneda

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.

Cite

@article{arxiv.2607.13025,
  title  = {The Balanced Four-Color Theorem},
  author = {Ken-ichi Kawarabayashi and Hirotaka Yoneda and Masataka Yoneda},
  journal= {arXiv preprint arXiv:2607.13025},
  year   = {2026}
}