{
  "abstract": "A conjecture of Sheehan states that every Hamiltonian $4$-regular graph contains a second Hamilton cycle. A natural problem is to find the smallest positive integer $k$ such that every Hamiltonian $r$-regular graph contains a second Hamilton cycle when $r\\geq k$. Thomassen proved that $k\\leq 300$. Improving Thomassen's result, Haxell, Seamone, and Verstra{ë}te proved that $k\\leq 23$. We improve this bound by proving that every $r$-regular Hamiltonian graph contains a second Hamilton cycle whenever $r\\geq 17$. Our new idea is to establish a transversal theorem for set systems on a cycle using Harris's results on the Moser--Tardos algorithm for Local Lemma. We then apply Thomassen's theorem on independent dominating sets to obtain a second Hamilton cycle. For $r\\ge 18$, we also give some algorithms to find such a transversal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06985",
  "author_total": 3,
  "authors": [
    {
      "name": "Haiyang Liu"
    },
    {
      "name": "Xu Liu"
    },
    {
      "name": "Bo Ning"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a7ypy6w9zc1x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06985",
  "fulltext_url": null,
  "id": "2a7ypy6w9zc1x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06985",
  "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/2611.06985",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a7ypy6w9zc1x.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.06985",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06985"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Every $r$-regular Hamiltonian graph with $r\\geq 17$ contains a second Hamilton cycle",
  "title_text": "Every r-regular Hamiltonian graph with r≥ 17 contains a second Hamilton cycle",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/2a7ypy6w9zc1x",
  "venue": "arXiv",
  "versions": []
}