{
  "abstract": "We determine the maximum number of triangles in $n$-vertex $k$-regular graphs and characterize all extremal graphs for all admissible orders when $k$ is odd and for an explicit range of orders when $k$ is even. For even $k$, we also give explicit lower and upper bounds for the remaining orders. As a consequence, we obtain a lower bound on the number of independent sets of size three in $k$-regular graphs and characterize all equality cases. Our exact results confirm the conjecture of Petingi and Rodríguez and the triangle case of a conjecture of Cambie, de Joannis de Verclos and Kang for all odd $k$ and the specified orders when $k$ is even. Moreover, for every even $k\\ge18$, we construct counterexamples to both conjectures and obtain a recursive characterization of all extremal graphs of order at least $Ck$, where $C$ is an absolute constant.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39461",
  "author_total": 5,
  "authors": [
    {
      "name": "Ya-Hong Chen"
    },
    {
      "name": "Shi-Cai Gong"
    },
    {
      "name": "Changhong Lu"
    },
    {
      "name": "Zhuojie Tu"
    },
    {
      "name": "Long-Tu Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckqnp68149mq6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39461",
  "fulltext_url": null,
  "id": "ckqnp68149mq6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39461",
  "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/2610.39461",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckqnp68149mq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39461",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39461"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The maximum number of triangles in regular graphs",
  "title_text": "The maximum number of triangles in regular graphs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/ckqnp68149mq6",
  "venue": "arXiv",
  "versions": []
}