{
  "abstract": "For a connected graph $G$, let $γ(G)$ and $γ_c(G)$ denote its domination number and connected domination number, respectively. Let $M(n,k)$ be the maximum of $γ_c(G)-γ(G)$ over all connected $n$-vertex graphs of minimum degree at least $k$. Alon proved that \\[ 2\\left\\lfloor\\frac{n}{k+1}\\right\\rfloor-O(1)\\le M(n,k)<\\frac{n}{k+1} \\bigl(\\log\\lceil\\log(k+1)\\rceil+3\\bigr). \\] He proposed a problem to determine or estimate $M(n,k)$ for $n-1\\ge k\\ge 3$, and particularly remarked that it would be interesting to close the $\\log\\log(k+1)$ gap between the upper and lower bounds and decide whether or not $M(n,k)=Θ\\bigl(\\frac{n}{k+1}\\bigr)$. We give an asymptotic answer to Alon's problem for sufficiently large $k$. More precisely, $M(k+1,k)=0$. For $n>k+1$, let \\[ ν=\\frac{n}{k+1} \\text{ and } Φ(x)= \\frac{1}{x\\log\\!\\frac{x}{x-1}} \\ (\\text{for }x>1). \\] As $k\\to\\infty$, uniformly over all integers $n>k+1$, \\[ M(n,k)= \\big(Φ(ν)+o(1)\\big) ν\\log\\log(k+1). \\]",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29623",
  "author_total": 2,
  "authors": [
    {
      "name": "Chengli Li"
    },
    {
      "name": "Bo Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4cmggk9c0a1t9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29623",
  "fulltext_url": null,
  "id": "4cmggk9c0a1t9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29623",
  "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": "W7214307644",
  "pdf_url": "https://arxiv.org/pdf/2609.29623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4cmggk9c0a1t9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Alon's problem concerning the difference between connected domination number and domination number",
  "title_text": "On Alon's problem concerning the difference between connected domination number and domination number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/4cmggk9c0a1t9",
  "venue": "arXiv",
  "versions": []
}