{
  "abstract": "Let $G$ be a graph. The circumference of $G$, denoted by $cir(G)$, is the length of a longest cycle in $G$, or zero if $G$ is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every $t>0$, there is a constant $A=A(t)>0$ such that every 2-connected $t$-tough graph of order $n$ has circumference at least $A\\log n$; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected $t$-tough graph $G$ of order $n$ satisfies $cir(G)\\ge \\lceil \\log_k((k-1)n+1)\\rceil$, where $k=\\lceil 1/t\\rceil+2$. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Briański, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12135",
  "author_total": 1,
  "authors": [
    {
      "name": "Songling Shan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jyb56pvxz376.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12135",
  "fulltext_url": null,
  "id": "1jyb56pvxz376",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12135",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213201469",
  "pdf_url": "https://arxiv.org/pdf/2609.12135",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jyb56pvxz376.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12135",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12135"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Long Cycles in 2-Connected Tough Graphs",
  "title_text": "Long Cycles in 2-Connected Tough Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/1jyb56pvxz376",
  "venue": "arXiv",
  "versions": []
}