{
  "abstract": "Let $G$ be a graph on $n$ vertices, and let $e_G(S,V(G)\\setminus S)$ be the number of edges with exactly one endpoint in $S$. The Cheeger constant and the restricted Cheeger constant of $G$, where $k\\ge1$ is real, are, respectively, \\[ h(G)=\\min_{\\substack{\\emptyset\\ne S\\subseteq V(G)\\\\|S|\\le \\frac{n}{2}}} \\frac{e_G(S,V(G)\\setminus S)}{|S|} \\text{ and } h_k(G)=\\min_{\\substack{\\emptyset\\ne S\\subseteq V(G) |S|\\le\\min\\{k,\\frac{n}{2}\\}}} \\frac{e_G(S,V(G)\\setminus S)}{|S|}. \\] The contraction clique number $\\ccl(G)$ is the largest integer $r$ such that $G$ contains the complete graph $K_r$ as a minor. Krivelevich and Nenadov [Complete minors in graphs without sparse cuts, Int. Math. Res. Not. IMRN 12 (2021) 8996--9015] proved that, for every fixed $\\eps>0$ and all sufficiently large $n$ and $d$, if $G$ is a graph on $n$ vertices with maximum degree at most $d$, then $h(G)\\ge\\eps d$ and $h_{\\eps n}(G)\\ge(\\frac{1}{2}+\\eps)d$ imply $\\ccl(G)=Ω_\\eps(\\sqrt{\\frac{nd}{\\log d}})$. They asked to determine if one can guarantee the same lower bound on $\\ccl(G)$ without the additional condition on $h_{\\eps n}(G)$. They showed that this is the case when $d$ is a constant. We answer this question affirmatively. For every $\\eps>0$, there are constants $β=β(\\eps)>0$ and $n_0=n_0(\\eps)$ such that, whenever $d\\ge 3$ is an integer, for every graph $G$ with $n\\ge n_0$ vertices and maximum degree at most $d$, if $h(G)\\ge\\eps d$, then $\\ccl(G)\\geβ\\sqrt{\\frac{nd}{\\log d}}$. The dependence of this lower bound on $n$ and $d$ is best possible up to a constant factor. As a corollary, a lower bound is derived for the contraction clique number of $d$-regular graphs for which the second largest eigenvalue is bounded away from $d$, compared to earlier $\\frac{d}{2}$. The proof combines spectral properties of graphs with an analysis of lazy random walks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15414",
  "author_total": 3,
  "authors": [
    {
      "name": "Chengli Li"
    },
    {
      "name": "Leyou Xu"
    },
    {
      "name": "Bo Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4e50wag3bp2er.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15414",
  "fulltext_url": null,
  "id": "4e50wag3bp2er",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15414",
  "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": "W7213295628",
  "pdf_url": "https://arxiv.org/pdf/2609.15414",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4e50wag3bp2er.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15414",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15414"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Large Complete Minors from a Cheeger Condition",
  "title_text": "Large Complete Minors from a Cheeger Condition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/4e50wag3bp2er",
  "venue": "arXiv",
  "versions": []
}