{
  "abstract": "Let $F$ be a graph on $m$ vertices with at least one edge, and let $0<p\\le1$. We study when prescribed subgraph counts across disjoint vertex classes characterize $p$-quasirandom graph sequences. For positive multiplicities $\\bm=(m_1,\\ldots,m_r)$ summing to $m$ and positive class proportions $\\balpha=(α_1,\\ldots,α_r)$ with total at most one, we first classify symmetrized repeated-class counts for $r\\ge3$: when $0<p<1$, they are forcing precisely if $e(F)>1$ or $\\balpha\\ne\\bm/m$. We also obtain a complete classification of labelled transversal counts: for \\rev{$0<p<1$, the only nonforcing profile is $F=K_2$ with class proportions $(1/2,1/2)$.} Together, these results settle conjectures of Janson and Sós [European J. Combin. 46 (2015)], Hatami, Hatami and Li [European J. Combin. 56 (2016)], and Janson [arXiv:2606.20186v1 (2026)]. The common ingredient is a balanced-count theorem for every graph with more than one edge, proved through a degree-sequence inequality, and its consequence of additive rigidity. Our arguments also determine the bad indices of the associated functional nullspaces and isolate the exceptional single-edge profiles. In particular, a non-divisibility theorem for grouped elementary symmetric polynomials excludes the higher-order obstructions in the repeated-class setting. In contrast, for two repeated classes and every $p\\in(0,1)$, we construct nonquasirandom graph sequences satisfying the crossing-triangle counts at proportions $(1/3,2/3)$, with limiting edge density strictly below $p$. These counterexamples answer the crossing-triangle question negatively and show why the three-class assumption cannot be omitted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28124",
  "author_total": 3,
  "authors": [
    {
      "name": "Guoqing Wang"
    },
    {
      "name": "Linlin Wang"
    },
    {
      "name": "Jun Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5a5942ps5191s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28124",
  "fulltext_url": null,
  "id": "5a5942ps5191s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28124",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223087388",
  "pdf_url": "https://arxiv.org/pdf/2610.28124",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5a5942ps5191s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.28124",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28124"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Quasirandomness from balanced and repeated subgraph counts",
  "title_text": "Quasirandomness from balanced and repeated subgraph counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5a5942ps5191s",
  "venue": "arXiv",
  "versions": []
}