{
  "abstract": "For a graph $G$ and a set $B\\subseteq V(G)$, the spread $\\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\\geq 0$ the parameter $\\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\\mathrm{sp}(B)\\leq k$. Caro, Lauri and Zarb asked for the minimum of $\\mathrm{sp}(G,k)$ over the maximal planar graphs of order $n$; we write $\\mathrm{MP}(n,δ,k)$ for this minimum over the maximal planar graphs of order $n$ and minimum degree $δ\\in \\{3,4,5\\}$. This manuscript is intended as an exposition of the subject of spread in the degree sequence of a graph $G$, with emphasis on maximal planar graphs. We apply the general lower bound of the companion paper \\cite{P1} to this class, together with further ideas, some based on Caro--West and Caro--Lauri--Zarb and some new. In particular we asymptotically determine the values of $\\mathrm{MP}(n,δ,k)$ for every pair $(δ,k)$ with $δ\\in \\{3,4,5\\}$ and $k\\geq 0$. This solution recovers the case $\\mathrm{MP}(n,δ,0)$, which is the repetition number $\\mathrm{rep}(G)$ introduced by Caro and West.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11395",
  "author_total": 3,
  "authors": [
    {
      "name": "Yair Caro"
    },
    {
      "name": "Riste Škrekovski"
    },
    {
      "name": "Christina Zarb"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62gb1x6ww5pm3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11395",
  "fulltext_url": null,
  "id": "62gb1x6ww5pm3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11395",
  "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": "W7221109576",
  "pdf_url": "https://arxiv.org/pdf/2610.11395",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62gb1x6ww5pm3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11395",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11395"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spreads of degrees in maximal planar graphs -- an exposition",
  "title_text": "Spreads of degrees in maximal planar graphs -- an exposition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/62gb1x6ww5pm3",
  "venue": "arXiv",
  "versions": []
}