{
  "abstract": "Let $f_{p+1}(n,m)$ denote the minimum number of $K_{p+1}$-saturating edges in an $n$-vertex $K_{p+1}$-free graph with $m$ edges. In this paper, we investigate the stability of this parameter near the Turán thresholds, with particular emphasis on the role of the minimum degree. We introduce the refined parameter $f_{p+1,p}(n,m,δ)$. It is defined for $K_{p+1}$-free graphs on $n$ vertices with $m$ edges, minimum degree $δ$, and containing at least a copy of $K_p$. For every fixed integer $p\\ge3$, we establish a stability result when the number of edges is $\\mathrm{ex}(n,K_p)+o(n^2)$, showing that \\[ f_{p+1,p} (n,m,δ)=\\left(\\frac{(p-2)^2}{2(p-1)^3}+o(1)\\right)n^2, \\ \\text{for }\\ m=\\mathrm{ex}(n,K_p)+o(n^2), \\ δ\\geq \\left(\\frac{p-2}{p-1}+o(1)\\right)n. \\] We further obtain a stability theorem in the regime where the number of edges is quadratically smaller than $\\operatorname{ex}(n,K_{p+1})$. More precisely, for $0 \\leq c< \\frac{1}{2p(3p-1)}$, \\[ f_{p+1,p} (n,m,δ)=\\left(\\frac{1}{2p}+o(1)\\right)n^2, \\ \\text{for }\\ m=\\operatorname{ex}(n,K_{p+1})-(c+o(1))n^2, \\ δ\\geq \\left(\\frac{p-1}{p}-2c+o(1)\\right)n. \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40696",
  "author_total": 2,
  "authors": [
    {
      "name": "Xianglei Chen"
    },
    {
      "name": "Changxiang He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0c64eb3hmy0gt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40696",
  "fulltext_url": null,
  "id": "0c64eb3hmy0gt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40696",
  "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/2610.40696",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0c64eb3hmy0gt.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.40696",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40696"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotic stability of clique-saturating edges near Turán thresholds",
  "title_text": "Asymptotic stability of clique-saturating edges near Turán thresholds",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/0c64eb3hmy0gt",
  "venue": "arXiv",
  "versions": []
}