{
  "abstract": "Let $k\\ge3$ and $1\\le \\ell\\le k-1$. We study balanced $k$-partitions of a graph for which the union of any $\\ell$ parts induces few edges. We show that every graph $G$ with $n$ vertices and $m$ edges admits a balanced partition $V_1,\\ldots,V_k$ such that \\begin{equation*} \\max_{\\substack{A\\in\\binom{[k]}{\\ell}}}e_G\\left(\\bigcup_{i\\in A}V_i\\right)\\le\\frac{\\ell^2}{k^2}m+\\frac{\\ell^2(k-\\ell)}{k^2}(n-1)+\\frac{\\ell(k-\\ell)}{k(k-1)}\\sqrt{\\left(\\binom{k}{\\ell}-1\\right)m}. \\end{equation*} In the case $\\ell=2$, our result confirms a conjecture of Bollobás and Scott in a stronger form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30927",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhanping Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/79b765cvf6ak1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30927",
  "fulltext_url": null,
  "id": "79b765cvf6ak1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30927",
  "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": "W7214594268",
  "pdf_url": "https://arxiv.org/pdf/2609.30927",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/79b765cvf6ak1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.30927",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30927"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounds for Unions of Several Parts in Balanced Graph Partitions",
  "title_text": "Bounds for Unions of Several Parts in Balanced Graph Partitions",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/79b765cvf6ak1",
  "venue": "arXiv",
  "versions": []
}