{
  "abstract": "For a graph $G$, we investigate sufficient degree conditions under which either $G$ or its complement $\\overline{G}$ is $k$-connected. Motivated by a theorem of Hellwig and Volkmann, we introduce the notion of complement connectivity admissible pairs and define the parameter $\\mathcal{C}(k)$ as the minimum degree threshold guaranteeing that either $G$ or $\\overline{G}$ is $k$-connected whenever $δ(G)\\ge\\mathcal{C}(k)$ and $Δ(G)\\le |V(G)|-1-\\mathcal{C}(k)$. We prove that $\\mathcal{C}(k)=2k-2$ for every $k\\ge2$, thereby determining the exact degree threshold. As an application, we derive a Nordhaus--Gaddum type product condition, showing that for $n\\ge4k-5$, the condition $δ(G)δ(\\overline{G}) \\ge (2k-3)(n-2k+2)$ guarantees that either $G$ or $\\overline{G}$ is $k$-connected. Finally, we show that no analogous product condition depending only on $k$ can exist.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14292",
  "author_total": 1,
  "authors": [
    {
      "name": "Arsen Berk Tekdaş"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3j5rea5w9z6k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14292",
  "fulltext_url": null,
  "id": "e3j5rea5w9z6k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14292",
  "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": "W7221813555",
  "pdf_url": "https://arxiv.org/pdf/2610.14292",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3j5rea5w9z6k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14292",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14292"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Exact Degree Threshold for Complement Connectivity",
  "title_text": "The Exact Degree Threshold for Complement Connectivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/e3j5rea5w9z6k",
  "venue": "arXiv",
  "versions": []
}