The Turán number of sparse spanning graphs

Noga Alon, Raphael Yuster

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For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Let δ(H)>0 and δ(H) denote the minimum degree and maximum degree of H, respectively. We prove that for all n sufficiently large, if H is any graph of order n with δ(H)≤n/40, then ex(n,H)=(n-12)+δ(H)-1. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case H=Cn, and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs. A counter-example to their more general conjecture concerning the extremal number of bounded degree spanning hypergraphs is also given.

Original languageEnglish
Pages (from-to)337-343
Number of pages7
JournalJournal of Combinatorial Theory. Series B
Issue number3
StatePublished - May 2013

Bibliographical note

Funding Information:
E-mail addresses: [email protected] (N. Alon), [email protected] (R. Yuster). 1 Research supported in part by an ERC advanced grant, by a USA–Israeli BSF grant, and by the Israeli I-Core program.


  • Packing
  • Spanning subgraph
  • Turan number

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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