{
  "abstract": "Let $G=(G_1,...,G_n)$ be a graph system on a common $n$-vertex set $V$. For $v\\in V$, set \\[ δ(v)=\\min_{i\\in[n]}d_{G_i}(v), \\qquad E_{\\cap}=\\bigcap_{i=1}^n E(G_i), \\] and let $d^c(u,v)$ denote the length of a shortest rainbow $u$--$v$ path. We study Hamiltonicity conditions imposed only on pairs at small rainbow distance. Suppose that every $G_i$ is $2$-connected and that \\[ δ(u),δ(v)<\\frac n2, \\qquad d^c(u,v)\\le2 \\quad\\Longrightarrow\\quad uv\\in E_{\\cap}. \\] Our main result shows that any system satisfying this local Fan-type condition but containing no rainbow Hamilton cycle must satisfy \\[ E_{\\cap}=\\varnothing, \\qquad \\sum_{v\\in V}δ(v)\\le\\frac{n^2-1}{2}. \\] Consequently, either $E_{\\cap}\\ne\\varnothing$ or $\\sum_{v\\in V}δ(v)\\ge\\lceil n^2/2\\rceil$ guarantees a rainbow Hamilton cycle. We further prove that the local Ore-type condition \\[ δ(u)+δ(v)<n, \\qquad d^c(u,v)\\le2 \\quad\\Longrightarrow\\quad uv\\in E_{\\cap} \\] already forces a rainbow Hamilton cycle. Along the way, the Fan-type hypothesis yields a rainbow Hamilton path and a rainbow $(n-1)$-cycle. For diagonal systems, our conditions reduce to the classical Fan and local Ore conditions. The proofs combine rainbow rotation--extension with a double-counting argument in an auxiliary digraph. Keywords: graph system; rainbow Hamilton cycle; rainbow distance; Fan-type condition; Ore-type condition; rotation--extension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13977",
  "author_total": 2,
  "authors": [
    {
      "name": "Ailian Chen"
    },
    {
      "name": "Hao Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a55tb696th004.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13977",
  "fulltext_url": null,
  "id": "a55tb696th004",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13977",
  "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": "W7221757540",
  "pdf_url": "https://arxiv.org/pdf/2610.13977",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a55tb696th004.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.13977",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13977"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems",
  "title_text": "Local Fan-type obstructions and Ore-type conditions for rainbow Hamiltonicity in graph systems",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/a55tb696th004",
  "venue": "arXiv",
  "versions": []
}