{
  "schema_version": 1,
  "problem_number": "GRAPH-049",
  "title": "Non-Word-Representable Single-Triangle Graphs with Word-Representable Line Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every odd integer m >= 5, let G_m be the Mycielski graph of the cycle C_m with an additional edge joining the shadow copies of two adjacent cycle vertices. We prove that G_m is connected, has exactly one triangle, and is a minimal non-word-representable graph under induced-subgraph containment, while its line graph is word-representable. In fact, G_m is 4-vertex-critical. This gives connected examples of clique number three for a follow-up question of Mozhui, Dwary and Krishna. The smallest member of the family has eleven vertices and twenty-one edges. The proof combines a restriction lemma, a one-vertex extension of their explicit line-graph orientation, and a general vertex-criticality lemma for a single shadow-edge Mycielski extension.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "word-representable graphs",
    "line graphs",
    "Mycielski graphs",
    "semi-transitive orientations",
    "triangle",
    "GRAPH-049"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.1",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.1",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/graph-049/",
  "pdf_url": "https://eulersolve.org/papers/graph-049/paper.pdf?v=6f6e52fb8dda",
  "doi": "10.5281/zenodo.23433839",
  "zenodo_record_url": "https://zenodo.org/records/23433839",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The older GRAPH-049 question was answered by Mozhui, Dwary and Krishna (2025); its resolution is not claimed here as new. This note proves the connected unique-triangle strengthening of their clique-three follow-up. The disconnected case is immediate by adjoining a disjoint triangle. No minimum-order result or answer to their all-triangle-free-root question is claimed. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute priority is claimed. Version 1.1 proves induced minimality and 4-vertex-criticality; this is not edge-deletion minimality.",
  "files": {
    "paper.pdf": {
      "sha256": "6f6e52fb8dda46b48a8d82110bef23d4b87306e7fc9faf5c5f5a9bff569dc42d"
    },
    "source.zip": {
      "sha256": "518b5fa6ae5d55cfcda3238ee454bb59d1c68420961fc3bdf072876f11a7d867"
    },
    "verification_report.md": {
      "sha256": "b9f573c1733381cff47ff5d61b4986d2fdd0d61db51160d56253b778ae295ce3"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}