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Minimizing the number of 5-cycles in graphs with given edge-density

Part of: Graph theory

Published online by Cambridge University Press:  09 October 2019

Patrick Bennett ,
Andrzej Dudek ,
Bernard Lidický  and
Oleg Pikhurko
Patrick Bennett
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Andrzej Dudek*
Affiliation:
Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Bernard Lidický
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, USA
Oleg Pikhurko
Affiliation:
Mathematics Institute and DIMAP, University of Warwick, Coventry CV4 7AL, UK
*
*Corresponding author. email: [email protected]
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Abstract

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least

$$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$
copies of C5. This bound is optimal, since a matching upper bound is given by the balanced complete k-partite graph. The proof is based on the flag algebras framework. We also provide a stability result. An SDP solver is not necessary to verify our proofs.

MSC classification

Primary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 1 , January 2020 , pp. 44 - 67
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Supported in part by Simons Foundation grant 426894.

‡

Supported in part by Simons Foundation grant 522400.

§

Supported in part by NSF grant DMS-1600390.

Supported in part by ERC grant 306493.

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