{
  "abstract": "Cycles $C_1,\\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\\subseteq\\cdots\\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02234",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiangdong Ai"
    },
    {
      "name": "Gregory Gutin"
    },
    {
      "name": "Yiming Hao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4k8wp25y3k3dw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02234",
  "fulltext_url": null,
  "id": "4k8wp25y3k3dw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02234",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7207862279",
  "pdf_url": "https://arxiv.org/pdf/2609.02234",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4k8wp25y3k3dw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.02234",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02234"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A linear bound for nested cycles without geometric crossings",
  "title_text": "A linear bound for nested cycles without geometric crossings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/4k8wp25y3k3dw",
  "venue": "arXiv",
  "versions": []
}