{
  "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 $k\\geq0$ the parameter $\\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\\mathrm{sp}(B)\\leq k$. Caro, Lauri and Zarb derived a lower bound for $\\mathrm{sp}(G,k)$ and, among several families of graphs, considered \\[ \\mathrm{MOP}(n,k)=\\min \\{\\mathrm{sp}(G,k):G\\text{ is a maximal outerplanar graph of order }n\\} \\] and determined $\\mathrm{MOP}(n,k)$ up to an additive constant for every $k\\not =2,$ leaving the case $k=2$ open, with the bounds $4n/9\\leq \\mathrm{MOP}(n,2)\\leq (5n+19)/11$. We first prove a lower bound on $\\mathrm{sp}(G,k)$ for an arbitrary graph $G$ in terms of its order $n$, its number of edges $m$ and its minimum degree $δ$. This lower bound contains the bounds of Caro, Lauri and Zarb and, for $k=0$, the bound $\\mathrm{rep}(G)\\geq \\left\\lceil n/(2d-2δ+1)\\right\\rceil $ of Caro and West, where $d=2m/n$. We determine when this lower bound is attained, exhibit explicit graphs attaining it, and show that it is exact for all graphs once $n\\geq n_{0}(δ,k,d)$. We then apply the bound to maximal outerplanar graphs: adjusting the count to this class we prove \\[ \\mathrm{MOP}(n,2)\\geq \\left\\lceil \\frac{4n+10}{9}\\right\\rceil \\qquad \\text{for every }n\\geq 14, \\] with equality for $n\\equiv 2\\ (\\mathrm{mod}\\ 18)$, and $\\mathrm{MOP}(n,2)=4n/9+O(1)$ for every $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19762",
  "author_total": 3,
  "authors": [
    {
      "name": "Yair Caro"
    },
    {
      "name": "Riste Škrekovski"
    },
    {
      "name": "Christina Zarb"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90ffjmvh19qkn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19762",
  "fulltext_url": null,
  "id": "90ffjmvh19qkn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19762",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213585430",
  "pdf_url": "https://arxiv.org/pdf/2609.19762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90ffjmvh19qkn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spreads of degrees in graphs",
  "title_text": "Spreads of degrees in graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/90ffjmvh19qkn",
  "venue": "arXiv",
  "versions": []
}