{
  "abstract": "We compare transversal and pattern thresholds for two spanning structures. For every fixed $k\\ge 3$ and $α>0$, we prove that every sufficiently large $n$-vertex $k$-graph collection with minimum codegree at least $(1/2+α)n$ contains a tight Hamilton cycle with any prescribed color pattern. Hence the transversal and pattern thresholds for tight Hamilton cycles coincide asymptotically. For spanning trees, we show that for every $α>0$ there exists $c=c(α)>0$ such that every $n$-vertex graph collection with minimum degree at least $(1/2+α)n$ contains every $n$-vertex tree with maximum degree at most $n^c$ and any prescribed color pattern. On the other hand, we construct spanning trees of maximum degree $\\sqrt n+1$ whose pattern threshold exceeds their transversal threshold by a positive linear term.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07421",
  "author_total": 1,
  "authors": [
    {
      "name": "Lirong Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/325sggp6z7vxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07421",
  "fulltext_url": null,
  "id": "325sggp6z7vxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07421",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/325sggp6z7vxm.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Prescribed Color Patterns in Tight Hamilton Cycles and Spanning Trees",
  "title_text": "Prescribed Color Patterns in Tight Hamilton Cycles and Spanning Trees",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/325sggp6z7vxm",
  "venue": "arXiv",
  "versions": []
}