{
  "abstract": "A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\\mathcal{O},$ of which only $\\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04956",
  "author_total": 3,
  "authors": [
    {
      "name": "Evangelos Protopapas"
    },
    {
      "name": "Dimitrios M. Thilikos"
    },
    {
      "name": "Sebastian Wiederrecht"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5zh5m62jbg3c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04956",
  "fulltext_url": null,
  "id": "c5zh5m62jbg3c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04956",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7211890435",
  "pdf_url": "https://arxiv.org/pdf/2609.04956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5zh5m62jbg3c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Erdős-Pósa Property for Colorful Minors",
  "title_text": "The Erdős-Pósa Property for Colorful Minors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/c5zh5m62jbg3c",
  "venue": "arXiv",
  "versions": []
}