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

  • Patrick Bennett (a1), Andrzej Dudek (a1), Bernard Lidický (a2) and Oleg Pikhurko (a3)

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.

Copyright

Corresponding author

*Corresponding author. email: andrzej.dudek@wmich.edu

Footnotes

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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.

Footnotes

References

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

  • Patrick Bennett (a1), Andrzej Dudek (a1), Bernard Lidický (a2) and Oleg Pikhurko (a3)

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