{
  "abstract": "A graph $G$ is said to be $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e$ of $\\overline{G}$. The saturation number of $H$, denoted by $\\mathrm{sat}(n, H)$, is the minimum number of edges among all $H$-saturated graphs of order $n$. In this paper, we investigate the saturation number of $K_3 \\vee P_k$ for $k \\ge 3$ and show that $\\mathrm{sat}(n, K_3 \\vee P_k) = 3n - 6 + \\mathrm{sat}(n-3, P_k)$ holds for all $n \\ge a_k + 3$ and $k \\ge 3$, which verifies the conjecture [Discrete Appl. Math, 394(2026) 62-80] on $\\mathrm{sat}(n, K_s \\vee P_k)$ is true for the case $s=3$, where $a_k = k$ for $3 \\le k \\le 5$, $a_k = 3 \\cdot 2^{t-1} - 2$ for $k = 2t \\ge 6$, and $a_k = 2^{t+1} - 2$ for $k = 2t + 1 \\ge 7$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34990",
  "author_total": 3,
  "authors": [
    {
      "name": "Huaqin Li"
    },
    {
      "name": "Lihua You"
    },
    {
      "name": "Xiaoxue Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bnzkgkn2g284g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34990",
  "fulltext_url": null,
  "id": "bnzkgkn2g284g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34990",
  "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": "W7223842708",
  "pdf_url": "https://arxiv.org/pdf/2610.34990",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bnzkgkn2g284g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34990",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34990"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Saturation numbers of $K_3 \\vee P_k$",
  "title_text": "Saturation numbers of K_3 ∨ P_k",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/bnzkgkn2g284g",
  "venue": "arXiv",
  "versions": []
}