{
  "abstract": "A graph with maximum degree $Δ$ is \\emph{$Δ$-critical} if it is connected, has chromatic index $Δ+1$, and the deletion of any edge reduces its chromatic index to $Δ$; it is \\emph{nontrivial} if it has no subgraph $H$ with $|E(H)|>Δ\\lfloor |V(H)|/2\\rfloor$. We prove a characterization of all nontrivial $3$-critical graphs: every such graph is obtained either from smaller $3$-critical graphs by replacing a vertex with a triangle, by a Hajós-join, or by replacing a vertex with a suitable class~$1$ graph, or from a snark by deleting at most one vertex and some independent edges. The third operation, which we call a Meredith-type extension, generalizes the Meredith extension by allowing other class~$1$ graphs in place of $K_{2,3}$, and we give a sufficient condition under which it produces new nontrivial $3$-critical graphs from smaller ones. To the best of our knowledge, this is the first characterization of this kind for edge-chromatic $3$-critical graphs, and the Meredith-type extension has not previously been considered in this generality. The characterization was found by analyzing the output of an exhaustive computer search for nontrivial $3$-critical graphs of orders $13$ to $24$, and we make the resulting data publicly available. Based on these data, we also propose two conjectures that are closely related to the case $Δ=3$ of Seymour's exact conjecture and to Grötzsch's conjecture on the $3$-edge-colorability of planar graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.27131",
  "author_total": 2,
  "authors": [
    {
      "name": "Le Chen"
    },
    {
      "name": "Songling Shan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8m0dqxzmrk3z7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.27131",
  "fulltext_url": null,
  "id": "8m0dqxzmrk3z7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.27131",
  "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": "W7166013706",
  "pdf_url": "https://arxiv.org/pdf/2606.27131",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8m0dqxzmrk3z7.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2606.27131",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.27131"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A characterization of edge-chromatic 3-critical graphs",
  "title_text": "A characterization of edge-chromatic 3-critical graphs",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:51Z",
  "url": "https://research.draconitic.com/paper/8m0dqxzmrk3z7",
  "venue": "arXiv",
  "versions": []
}