{
  "abstract": "The Total Coloring Conjecture (TCC) is a challenging unsolved problem posed by Behzad and Vizing independently, which states that every simple graph $G$ admits a ($Δ(G)$ +2)-total-coloring, where $Δ(G)$ denotes the maximum degree of $G$. This conjecture has been confirmed for graphs with $Δ(G)\\leq 5$. However, for planar graphs, the only open case is $Δ(G)=6$. It was known that planar graphs with maximum degree 6 and without 4-cycles are 7-totally-colorable. In this paper, we improve this result by showing that any planar graph $G$ of maximum degree 6, which does not contain some special 4-cycles, is 7-totally-colorable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19503",
  "author_total": 3,
  "authors": [
    {
      "name": "Enqiang Zhu"
    },
    {
      "name": "Yangyang Zhou"
    },
    {
      "name": "Jin Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d83mnfbr3ha92.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19503",
  "fulltext_url": null,
  "id": "d83mnfbr3ha92",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19503",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Graph Labeling and Dimension Problems"
    ]
  ],
  "openalex": "W7213591991",
  "pdf_url": "https://arxiv.org/pdf/2609.19503",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d83mnfbr3ha92.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19503",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19503"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles",
  "title_text": "Total-coloring of planar graphs with maximum degree 6 and without prescribed 4-cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/d83mnfbr3ha92",
  "venue": "arXiv",
  "versions": []
}