{
  "abstract": "A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\\le 3$ and remains open for all $k\\ge 4$. In this paper, we prove Bondy's conjecture for all sufficiently large graphs. The key ingredient is a new Dirac-type theorem that gives a lower bound on the length of a longest cycle in a $k$-connected graph, which also yields a partial solution to a conjecture of Jung from 1990. Along the way, we develop several new tools, including a DFS lemma and an average-degree analogue of the Bondy--Jackson theorem. We conclude with a discussion of related problems and a counterexample to a conjecture of Voss from 1991.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: International Mathematics Research Notices 2026(21)",
  "arxiv": "2606.03696",
  "author_total": 3,
  "authors": [
    {
      "name": "Jie Ma"
    },
    {
      "name": "Bo Ning"
    },
    {
      "name": "Ziyuan Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6cdpr54y465vx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.03696",
  "fulltext_url": null,
  "id": "6cdpr54y465vx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.03696",
  "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/2606.03696",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6cdpr54y465vx.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:33Z",
      "native_id": "2606.03696",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.03696"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Longest cycles and Dirac-type results in highly connected graphs",
  "title_text": "Longest cycles and Dirac-type results in highly connected graphs",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:33Z",
  "url": "https://research.draconitic.com/paper/6cdpr54y465vx",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1093/imrn/rnag249",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}