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arXiv · 2510.15407

5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$

Yuta Inoue·Ken-ichi Kawarabayashi·Atsuyuki Miyashita
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Abstract

We show that any planar graph $G=(V,E)$ has a 5-coloring such that one color class contains at most $|V|/6$ vertices. In other words, there exists a partition of $V$ into five independent sets $\{V_1, \cdots, V_5\}$ such that $|V_5| \leq |V| / 6$. Our proof yields an $O(|V|^2)$-time algorithm to find such a partition, and unlike the Four Color Theorem, our proof is fully verifiable without computer assistance.

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Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita. 2025-10-17. 5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$. https://arxiv.org/abs/2510.15407

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