{
  "abstract": "The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\\log n)$ amortized time. Thus, we implement all $O(\\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39753",
  "author_total": 6,
  "authors": [
    {
      "name": "Yuta Inoue"
    },
    {
      "name": "Ken-ichi Kawarabayashi"
    },
    {
      "name": "Atsuyuki Miyashita"
    },
    {
      "name": "Bojan Mohar"
    },
    {
      "name": "Carsten Thomassen"
    },
    {
      "name": "Mikkel Thorup"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5smg3gen4k2yy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39753",
  "fulltext_url": null,
  "id": "5smg3gen4k2yy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39753",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39753",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5smg3gen4k2yy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39753",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39753"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Four-Coloring Planar Graphs in Linear Time",
  "title_text": "Four-Coloring Planar Graphs in Linear Time",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/5smg3gen4k2yy",
  "venue": "arXiv",
  "versions": []
}