5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912654410383360 |
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| author | Inoue, Yuta Kawarabayashi, Ken-ichi Miyashita, Atsuyuki |
| author_facet | Inoue, Yuta Kawarabayashi, Ken-ichi Miyashita, Atsuyuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_15407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | 5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$ Inoue, Yuta Kawarabayashi, Ken-ichi Miyashita, Atsuyuki Combinatorics 05C15 (Primary) 05C10, 05C85 (Secondary) 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. |
| title | 5-Coloring Planar Graphs with a Color Class of Order at Most $|V|/6$ |
| topic | Combinatorics 05C15 (Primary) 05C10, 05C85 (Secondary) |
| url | https://arxiv.org/abs/2510.15407 |