{
  "abstract": "For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for every possible chromatic number. Let $T_{n,r}$ be the complete $r$-partite Turán graph on $n$ vertices. We find a constant $γ_r>0$ such that if $n$ is sufficiently large and $\\mathcal{H}$ is a finite family with $χ(\\mathcal{H})=r+1$ satisfying \\[ ex(n,\\mathcal{H})<e(T_{n,r})+γ_r n, \\] then every cycle-maximal $\\mathcal{H}$-free graph is edge-extremal, and $γ_r$ is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12109",
  "author_total": 3,
  "authors": [
    {
      "name": "Jifu Lin"
    },
    {
      "name": "Xiaolin Wang"
    },
    {
      "name": "Guangmiao Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bc9qqnwvcq5e8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12109",
  "fulltext_url": null,
  "id": "bc9qqnwvcq5e8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12109",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221173474",
  "pdf_url": "https://arxiv.org/pdf/2610.12109",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bc9qqnwvcq5e8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12109",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12109"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp linear Turán remainders for cycle and edge extremality",
  "title_text": "Sharp linear Turán remainders for cycle and edge extremality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/bc9qqnwvcq5e8",
  "venue": "arXiv",
  "versions": []
}