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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)



The number of cliques in graphs of given order and size

Author: V. Nikiforov
Journal: Trans. Amer. Math. Soc. 363 (2011), 1599-1618
MSC (2010): Primary 05C35
Published electronically: October 22, 2010
MathSciNet review: 2737279
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Abstract: Let $ k_{r}\left(n,m\right) $ denote the minimum number of $ r$-cliques in graphs with $ n$ vertices and $ m$ edges. For $ r=3,4$ we give a lower bound on $ k_{r}\left(n,m\right) $ that approximates $ k_{r}\left(n,m\right) $ with an error smaller than $ n^{r}/\left(n^{2}-2m\right).$

The solution is based on a constraint minimization of certain multilinear forms. Our proof combines a combinatorial strategy with extensive analytical arguments.

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Additional Information

V. Nikiforov
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152

Received by editor(s): April 5, 2009
Received by editor(s) in revised form: August 18, 2009
Published electronically: October 22, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.