{
  "abstract": "An oddomorphism to $K_n$ is a proper colouring with $n$ colours in which every vertex is designated odd or even, every vertex has in each other colour class a number of neighbours of its own parity, and every colour class contains an odd number of odd vertices. We prove that an oddomorphism witness to $K_n$ on at most $n+6$ vertices contains a $K_n$ minor with every branch meeting the witness's specified odd set. The conclusion holds inside any graph containing the witness and therefore applies to weak oddomorphisms. The bound is sharp: for every $n\\ge8$ the minimum order of a graph admitting a weak oddomorphism to $K_n$ but having no $K_n$ minor is exactly $n+7$. The extremal example is a fifteen-vertex graph with an oddomorphism to $K_8$, obtained from the doubled theta graph of Kriesell and Mohr by adding one vertex adjacent to the seven roots; we give a structural proof that it has no $K_8$ minor, including their counting argument for the rooted case. A separate technical companion records further results on related root-apex graphs, an attachment theorem and a bounded nine-colour statement, together with their computational provenance; none of it is used here. The results do not resolve the unrestricted clique-target problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33283",
  "author_total": 1,
  "authors": [
    {
      "name": "Michal Čertík"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/57m1cyndv5d4b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33283",
  "fulltext_url": null,
  "id": "57m1cyndv5d4b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33283",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223818484",
  "pdf_url": "https://arxiv.org/pdf/2610.33283",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/57m1cyndv5d4b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33283",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33283"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Minimum orders of clique oddomorphism counterexamples",
  "title_text": "Minimum orders of clique oddomorphism counterexamples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/57m1cyndv5d4b",
  "venue": "arXiv",
  "versions": []
}