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Combinatorica

, Volume 2, Issue 3, pp 275–288 | Cite as

Compactness results in extremal graph theory

  • P. Erdős
  • M. Simonovits
Article

Abstract

Let L be a given family of so called prohibited graphs. Let ex (n, L) denote the maximum number of edges a simple graph of ordern can have without containing subgraphs from L. A typical extremal graph problem is to determine ex (n, L), or at least, find good bounds on it. Results asserting that for a given L there exists a much smaller L*⫅L for which ex (n, L) ≈ ex (n, L*) will be calledcompactness results. The main purpose of this paper is to prove some compactness results for the case when L consists of cycles. One of our main tools will be finding lower bounds on the number of pathsP k+1 in a graph ofn vertices andE edges., witch is, in fact, a “super-saturated” version of a wellknown theorem of Erdős and Gallai.

AMS subject classification (1980)

05 C 35 

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Copyright information

© Akadémiai Kiadó 1982

Authors and Affiliations

  • P. Erdős
    • 1
  • M. Simonovits
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.Dept. Analysis I.Eötvös UniversityBudapestHungary

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