{
  "abstract": "An edge-colored graph is \\emph{rainbow} if all of its edges receive distinct colors. For a fixed graph $H$, an edge-colored graph $F$ is called weakly $H$-rainbow saturated if there exists an ordering $e_1,e_2,\\ldots,e_{|E(\\bar{F})|}$ of $E(\\bar{F})$ such that, for any edge coloring $c$ of $E(\\bar{F})$ with $c(e_i)\\neq c(e_j)$, there is always a rainbow copy of $H$ that contains $e_i$ in $F+\\{e_1,e_2,\\ldots,e_i\\}$. The \\emph{weak rainbow saturation number} $\\operatorname{rwsat}(n,H)$ is the minimum number of edges in a weakly $H$-rainbow saturated graph on $n$ vertices. Li, Ma, and Xie [JGT, 2025] showed that $\\lim_{n\\to\\infty} \\frac{\\operatorname{rwsat}(n,H)}{n}$ exists for every nonempty graph $H$. Paths and stars attain, respectively, the minimum and maximum ordinary weak saturation numbers among all trees of the same order. We determine their weak rainbow saturation numbers exactly. For all $\\ell>30$, we show that $$ \\ell+1=\\s(n,P_\\ell)< \\s(n,S_\\ell)=\\binom{\\ell}{2}-1$$ where $P_\\ell$ and $S_\\ell$ denote the path and star on $\\ell$ vertices, respectively. Thus, their dependence on $\\ell$ is linear for paths and quadratic for stars. We then focus on cycles. Li, Ma, and Xie asked whether $\\operatorname{rwsat}(n,C_\\ell)$ has leading term $\\frac32n$ for every $\\ell\\ge4$. We answer this question negatively by giving an explicit construction showing that, for every $\\ell\\ge4$ and all sufficiently large $n$, $$\\s(n,C_\\ell)< \\frac{\\ell}{\\ell-1}n+c_\\ell,$$ where $c_\\ell$ depends only on $\\ell$. Since $\\frac{\\ell}{\\ell-1}<\\frac32$, this strictly improves the proposed leading coefficient for every cycle $C_\\ell$ with $\\ell\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03823",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiawen Bo"
    },
    {
      "name": "Xiaopan Lian"
    },
    {
      "name": "Jianing Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bnjp7b1nzx3dz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03823",
  "fulltext_url": null,
  "id": "bnjp7b1nzx3dz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03823",
  "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": "W7208716161",
  "pdf_url": "https://arxiv.org/pdf/2609.03823",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bnjp7b1nzx3dz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03823",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03823"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Weak rainbow saturation numbers of paths, stars and cycles",
  "title_text": "Weak rainbow saturation numbers of paths, stars and cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/bnjp7b1nzx3dz",
  "venue": "arXiv",
  "versions": []
}