{
  "abstract": "Let $\\mathbf{G}=\\{G_1,\\dots,G_{n-1}\\}$ be a collection of not necessarily distinct graphs of order $n$ on the same vertex set $V$. A path $P$ with $V(P)\\subseteq V$ is called \\emph{rainbow} in $\\mathbf{G}$, if there exists an injection $ϕ\\colon E(P)\\to [n-1]$ such that $e\\in E(G_{ϕ(e)})$ for each $e\\in E(P)$. In this paper, we show that if $n\\ge3$, $e(G_i)> \\binom{n-2}{2}+2$ for each $i\\in[n-1]$ and $G_j$ is connected for some $j\\in[n-1]$, then there exists a rainbow Hamiltonian path in $\\mathbf{G}$ unless $n=6$ with $G_1=G_2=\\dots=G_{5}\\cong K_2\\vee 4K_1$, or $n=8$ with $G_1=G_2=\\dots=G_{7}\\cong K_3\\vee 5K_1$. This corrects Theorem 6 in [Graphs Combin., \\textbf{41}(6)(2025), 119] and relaxes its hypothesis from requiring all $G_i$ to be connected to only requiring that at least one $G_i$ is connected, where $i\\in[n-1]$. Moreover, we show that if $n\\ge2$ and $e(G_i)\\ge \\binom{n-1}{2}$ for each $i\\in[n-1]$, then $\\mathbf{G}$ contains a rainbow Hamiltonian path unless $G_1=G_2=\\dots=G_{n-1}\\cong K_{n-1}\\cup K_1$, or $n=4$ with $G_1=G_2=G_3\\cong K_{1,3}$. As a consequence, Theorem 2 of [Discrete Math., \\textbf{347}(2024), 114128] follows immediately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41203",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaoxue Zhang"
    },
    {
      "name": "Lihua You"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fswnx5ra1cc3f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41203",
  "fulltext_url": null,
  "id": "fswnx5ra1cc3f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41203",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fswnx5ra1cc3f.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Rainbow Hamiltonian paths in graph collections under edge-size conditions",
  "title_text": "Rainbow Hamiltonian paths in graph collections under edge-size conditions",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/fswnx5ra1cc3f",
  "venue": "arXiv",
  "versions": []
}