{
  "abstract": "Let $F$ be a fixed graph without isolated vertices. An edge-colored graph is $F$-rainbow saturated if it contains no rainbow copy of $F$, but the addition of any missing edge in any color creates a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in such a graph on $n$ vertices. We prove a dichotomy governed by isolated edges: if $F$ contains an isolated edge, then $rsat(n,F)=O(1)$ for all sufficiently large $n$, while if $F$ has no isolated edge, then $rsat(n,F)=Θ(n)$. The linear lower bound is expressed in terms of a directed weight parameter $η(F)$ and establishes the linear half of the dichotomy; in several cases it also strengthens the Cameron--Puleo type coefficient. For the bounded half, we construct rainbow saturated graphs for targets of the form $H\\cup K_2$. As an application of these constructions, we determine the asymptotically tight behavior for the rainbow saturation number of the generalized friendship graph $F_{t,p,q}=tK_p\\vee K_q$, proving that $ rsat(n,F_{t,p,q})=(p+q-1)n+O(1)$ for fixed $t\\geq 2$, $p\\geq 2$ and $q\\geq 1$ as $n\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34898",
  "author_total": 4,
  "authors": [
    {
      "name": "Yanzhe Qiu"
    },
    {
      "name": "Mei Lu"
    },
    {
      "name": "Yilin Pan"
    },
    {
      "name": "Yiduo Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ba7xkv142wa4z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34898",
  "fulltext_url": null,
  "id": "ba7xkv142wa4z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34898",
  "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": "W7214782185",
  "pdf_url": "https://arxiv.org/pdf/2609.34898",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ba7xkv142wa4z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34898",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34898"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On rainbow saturated graphs with minimum number of edges",
  "title_text": "On rainbow saturated graphs with minimum number of edges",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ba7xkv142wa4z",
  "venue": "arXiv",
  "versions": []
}