{
  "abstract": "We show that every $K_h$-minor-free graph with $n$ vertices has a balanced separator of size $O(h\\sqrt{n})$ for any parameter $h \\in \\mathbb{N}$, resolving the conjecture posed by Alon, Seymour, and Thomas [AST90] in 1990. The proof is elementary, building on the combinatorial framework by Le [Le26]. A key new idea of the proof is the existence of a short hitting walk on vertex expanders: given $k$ large (not necessarily disjoint) subsets of vertices, there is a walk of length $O(k/ψ(G))$ hitting every subset, where $ψ(G)$ is the vertex expansion. This technical result substantially improves on an analogous lemma by Alon, Seymour, and Thomas [AST90] in an important parameter setting, where the previous lower bound [Kot00] does not apply.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00663",
  "author_total": 3,
  "authors": [
    {
      "name": "Jonathan Conroy"
    },
    {
      "name": "Hung Le"
    },
    {
      "name": "Jack Spalding-Jamieson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3086vfc3ye2p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00663",
  "fulltext_url": null,
  "id": "e3086vfc3ye2p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00663",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00663",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3086vfc3ye2p.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00663",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00663"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$K_h$-Minor-Free Graphs Have Balanced Separators of Size $O(h\\sqrt{n})$",
  "title_text": "K_h-Minor-Free Graphs Have Balanced Separators of Size O(h√n)",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/e3086vfc3ye2p",
  "venue": "arXiv",
  "versions": []
}