{
  "abstract": "For a graph $G$, we write $\\mathrm{mad}(G)$ for its maximum average degree and $\\mathrm{diam} G$ for its diameter. Let $R_k(G)$ be the graph whose vertices are the proper colorings of $G$ with $k$ colors, where two colorings are adjacent when they differ at one vertex. Feghali (JCTB, 2021) proved that, for fixed integers $d,k\\ge 1$ with $k\\ge d+1$ and every $\\varepsilon>0$, every $n$-vertex graph $G$ satisfying $\\mathrm{mad}(G)\\le d-\\varepsilon$ has $\\mathrm{diam} R_k(G)=O_{d,k,\\varepsilon}(n(\\log n)^{d-1})$. In this article, we prove that \\[ \\mathrm{diam} R_k(G)=O_{d,k,\\varepsilon}\\!\\left( n(\\log n)^{\\left\\lfloor (d-1)/(k-d)\\right\\rfloor} \\right), \\] which extends the result proved by Feghali directly. The proof uses a partition into independent layers and removes $k-d$ colors at each recursive stage. We also improve the bound on the number of layers and determine the best possible linear coefficient in the forest case. More precisely, for $0<\\varepsilon<2$, every $n$-vertex graph $G$ with $\\mathrm{mad}(G)\\le2-\\varepsilon$ satisfies $\\mathrm{diam} R_3(G)\\leρ_M n$, where $M=\\lfloor2/\\varepsilon\\rfloor$, $ρ_M=\\max_{1\\le m\\le M}D_m/m$, and $D_m$ is the largest diameter of $R_3(T)$ over all trees $T$ on $m$ vertices. Moreover, the coefficient $ρ_M$ is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32688",
  "author_total": 2,
  "authors": [
    {
      "name": "Ruilin Zheng"
    },
    {
      "name": "Junying Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/64m6j2mw594g0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32688",
  "fulltext_url": null,
  "id": "64m6j2mw594g0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32688",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214722224",
  "pdf_url": "https://arxiv.org/pdf/2609.32688",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/64m6j2mw594g0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32688",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32688"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The diameter of recoloring graphs under a maximum average degree bound",
  "title_text": "The diameter of recoloring graphs under a maximum average degree bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/64m6j2mw594g0",
  "venue": "arXiv",
  "versions": []
}