{
  "abstract": "For a finite graph $G$, the maximum average degree $\\operatorname{mad}(G)$ is the largest average degree of a nonempty subgraph of $G$. Hendrey, Norin and Wood asked whether this parameter is partitionable, that is, whether for all positive reals $a,b$ every graph $G$ with $\\operatorname{mad}(G)<a+b$ admits a vertex partition $V(G)=A\\cup B$ with $\\operatorname{mad}(G[A])<a$ and $\\operatorname{mad}(G[B])<b$. In this note, we answer this question affirmatively. Moreover, we generalize this to an arbitrary number of partition classes. The clustered chromatic number $χ_\\star(\\mathrm{G})$ of a graph class $\\mathrm{G}$ is the minimum integer $k$ such that, for some integer $c$, every graph in $\\mathrm{G}$ has a $k$-coloring in which every monochromatic component has at most $c$ vertices. We apply this result to show that $χ_\\star(\\mathrm{A}_m)=\\left\\lfloor\\frac{m}{2}\\right\\rfloor+1$, where $\\mathrm{A}_m$ is the family of graphs $G$ with $\\operatorname{mad}(G)\\leq m$. This solves an open problem posed in Wood's survey and highlighted by Hendrey and Wood. As another application of our general partition result for $\\operatorname{mad}$, we obtain a bound for relaxed colorings. Namely, for non-negative integers $d_1,\\ldots,d_k$, every graph $G$ with \\[\\operatorname{mad}(G)<\\sum_{i=1}^k \\frac{2d_i+2}{d_i+2}\\] admits a partition $V(G)=V_1\\cup\\ldots\\cup V_k$ such that $Δ(G[V_i])\\leq d_i$ for each $i\\in[k]$. In particular, this provides the first non-trivial bounds for $k\\ge3$ with arbitrary $d_i$ and improves the previously known bound for $(d+1,d)$-colorings with $d \\ge 2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04645",
  "author_total": 5,
  "authors": [
    {
      "name": "Andrzej Grzesik"
    },
    {
      "name": "Lenka Kopfová"
    },
    {
      "name": "Gaurav Kucheriya"
    },
    {
      "name": "Binlong Li"
    },
    {
      "name": "Magdalena Prorok"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/drt963ph79wq6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04645",
  "fulltext_url": null,
  "id": "drt963ph79wq6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04645",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220681263",
  "pdf_url": "https://arxiv.org/pdf/2610.04645",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/drt963ph79wq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.04645",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04645"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Proof of the mad conjecture and its coloring applications",
  "title_text": "Proof of the mad conjecture and its coloring applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/drt963ph79wq6",
  "venue": "arXiv",
  "versions": []
}