{
  "abstract": "In this paper we introduce and explore the following new graph invariant: For a graph $G$ on $k$ vertices, $G \\neq K_k$, let $st(n,G)$ denote the maximum number of edges in a graph of order $n$ which does not contain any subgraph on $k$ vertices strictly containing $G$. A basic relation to classical Turán numbers is developed via the following: For $G$ on $k$ vertices, let $D(G) = \\{ H : |H| = |G|, H = G + e \\}$. Using this notion we prove that $ex(n,G) \\leq st(n,G) = ex(n, D(G) ) \\leq \\min \\{ ex(n,H) : H \\in D(G) \\}$ holds for all $n \\geq |G|$. The family $D(G)$ happened to be smoothly amenable to the use of classical extremal results, and in many cases allows us to get asymptotically sharp estimates as well as exact values of $st(n,G)$. From the many results proved here we state the following as an illustration. (1) If $χ(G) \\geq 3$ and $χ(D(G)) = χ(G)$, then $st(n,G) = (1+o(1))ex(n,K_{χ(G)})$. (2) If $χ(D(G)) = χ(G) +1$, then $G$ is a complete $χ(G)$-partite graph and $st(n,G) = ex(n,K_{χ(G) +1})$ for $n$ sufficiently large. (3) For $k$ odd, $k\\geq 5$, $st(n,C_k) = ex(n,C_k) = ex(n,K_3)$ for $n$ sufficiently large. (4) If $T$ is a tree of order $q$ with diameter $k \\geq 2$ and $q \\geq k+1 \\geq 3$, then $ex(n, \\{C_3,...,C_{k+1}\\}) \\leq st(n,T) \\leq ex(n, \\{C_3,...,C_{k+1}\\}) + (q-1)n$. Many results concerning even cycles, theta graphs, dense bipartite graphs and graphs of the form $G = G^* \\cup tK_1$ are obtained, moreover the value of $st(n,G)$ is computed for all graphs on at most 4 vertices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29304",
  "author_total": 2,
  "authors": [
    {
      "name": "Yair Caro"
    },
    {
      "name": "Zsolt Tuza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwphncy8za65z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29304",
  "fulltext_url": null,
  "id": "fwphncy8za65z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29304",
  "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": "W7214382006",
  "pdf_url": "https://arxiv.org/pdf/2609.29304",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwphncy8za65z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29304",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29304"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The strong (non-induced) Turán numbers",
  "title_text": "The strong (non-induced) Turán numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/fwphncy8za65z",
  "venue": "arXiv",
  "versions": []
}