{
  "abstract": "A path in a properly edge-colored graph is rainbow if its edges have pairwise distinct colors. For a proper edge-coloring $c$ of a graph $G$, let $\\operatorname{rpc}(G,c)$ be the minimum number of rainbow paths needed to cover $E(G)$, and let $\\operatorname{rpc}(G)$ be the maximum of $\\operatorname{rpc}(G,c)$ over all proper edge-colorings of $G$. We prove that, for every fixed $0<α<1$, every properly edge-colored $n$-vertex graph with minimum degree at least $αn$ satisfies $\\operatorname{rpc}(G,c)\\leq(1+o(1))n/2$, where the coefficient $1/2$ is best possible. We also determine $\\operatorname{rpc}(G)$ asymptotically for every complete multipartite graph. If $G=K_{n_1,\\ldots,n_r}$ has order $n$ and largest and smallest part sizes $M$ and $s$, respectively, then, uniformly over all choices of the number and sizes of the parts, $\\operatorname{rpc}(G)=(1+o(1))\\max\\{\\min\\{\\lfloor n/2\\rfloor,n-M\\},(n-s)/2\\}$. The proof combines pseudorandom packings of globally rainbow linear forests with a decomposition into dense parts and prescribed avoidance for arbitrary dense graphs, and with reserved connectors and a direct dominant-part argument for complete multipartite graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18740",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiao-Chuan Liu"
    },
    {
      "name": "Boyan Xu"
    },
    {
      "name": "Xu Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vwtv2b4j8z50.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18740",
  "fulltext_url": null,
  "id": "7vwtv2b4j8z50",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18740",
  "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": "W7213523234",
  "pdf_url": "https://arxiv.org/pdf/2609.18740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vwtv2b4j8z50.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp Rainbow Path Covers in Dense and Complete Multipartite Graphs",
  "title_text": "Sharp Rainbow Path Covers in Dense and Complete Multipartite Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/7vwtv2b4j8z50",
  "venue": "arXiv",
  "versions": []
}