{
  "abstract": "For a vertex $v$ of a graph $G$, the triangle-degree $\\operatorname{td}_G(v)$ is the number of triangles containing $v$. A graph is triangle-distinct, or $K_3$-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erdős, and Oellermann asked whether a regular $K_3$-irregular graph exists. We prove that such graphs exist for every regularity $r\\ge 9$. More precisely, for every integer $k\\ge 15$ we construct a $2k$-regular triangle-distinct graph on $4k+2$ vertices; complementation gives a $(2k+1)$-regular example of the same order. The construction uses two antiregular threshold blocks joined by a zero-one matrix with prescribed margins, followed by matrix $2$-switches that preserve those margins. After a uniform reference perturbation, exactly three triangle-degree collisions remain. Switches chosen according to parity remove two of them, and the last possible collision is controlled by a short quadratic discriminant argument. Odd $k\\ge 17$ and even $k\\ge 62$ are handled symbolically, while the remaining 24 values are settled by exact finite verification. Together with the known examples for $9\\le r\\le 29$, this closes the positive existence problem for every $r\\ge 9$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08478",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhanhe Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cemmv8rsqqamv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08478",
  "fulltext_url": null,
  "id": "cemmv8rsqqamv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08478",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.08478",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cemmv8rsqqamv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.08478",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08478"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Regular $K_3$-Irregular Graphs of Every Regularity at Least Nine",
  "title_text": "Regular K_3-Irregular Graphs of Every Regularity at Least Nine",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cemmv8rsqqamv",
  "venue": "arXiv",
  "versions": []
}