{
  "abstract": "In 1984, Roland Häggkvist posed the problem of constructing Hamiltonian graphs of order $n$ with large minimum degree and no $(n-2)$-cycle. He remarked that he did not know of such a graph with minimum degree at least three. We solve this problem by proving the following two results. (1) For every integer $d\\ge 3$ and every integer $n\\ge 15d-14,$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-2)$-cycle. (2) For every integer $d\\ge 3,$ every positive integer $k,$ and every integer $n\\ge (k+1)[(d-1)(k+3)+1],$ there exists a Hamiltonian graph of order $n$ and minimum degree $d$ that contains no $(n-s)$-cycle for any $s\\in\\{1,2,\\dots,k\\}.$ The proofs are constructive. We also pose several open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05468",
  "author_total": 1,
  "authors": [
    {
      "name": "Xingzhi Zhan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9gxb3nz4k9z95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05468",
  "fulltext_url": null,
  "id": "9gxb3nz4k9z95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05468",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212046248",
  "pdf_url": "https://arxiv.org/pdf/2609.05468",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9gxb3nz4k9z95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05468",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05468"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles",
  "title_text": "Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/9gxb3nz4k9z95",
  "venue": "arXiv",
  "versions": []
}