{
  "abstract": "For a graph $F$, its extremal number ${\\rm ex}(n,F)$ and extremal graphs denote the maximum number of edges in $F$-free graphs and the graphs attaining this maximum, respectively. Let $h_F(n,q)$ be the minimum number of copies of $F$ in an $n$-vertex graph with ${\\rm ex}(n,F)+q$ edges, and let $t_F(n,q)$ denote the corresponding minimum obtained by adding $q$ edges to an extremal graph. Mubayi (2013) conjectured that for every non-$r$-partite stable $r$-uniform hypergraph $F$ with $r\\ge 2$, $h_F(n,q)\\ge q\\cdot t_F(n,1)$ holds for all sufficiently large $n$ and all integers $q\\ge 1$. In this paper, we prove that Mubayi's conjecture fails in a stronger sense. For every integer $d\\ge 2$, we construct a family of non-$d$-partite stable graphs $F$ such that $h_F(n,q)< q\\cdot t_F(n,1)$ for all $1\\le q\\le O\\bigl(n^{4/ 3}\\bigr)$. In particular, these examplesis completely refute Mubayi's conjecture. \\noindent\\emph{Key words}: extremal graph; supersaturation; chromatic number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18876",
  "author_total": 2,
  "authors": [
    {
      "name": "Liping Zhang"
    },
    {
      "name": "Qinghai Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nstza2vtne0g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18876",
  "fulltext_url": null,
  "id": "3nstza2vtne0g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18876",
  "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": "W7222108409",
  "pdf_url": "https://arxiv.org/pdf/2610.18876",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nstza2vtne0g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.18876",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18876"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Complete counterexamples to Mubayi's supersaturation conjecture for graphs",
  "title_text": "Complete counterexamples to Mubayi's supersaturation conjecture for graphs",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/3nstza2vtne0g",
  "venue": "arXiv",
  "versions": []
}