Minimizing the number of 5-cycles in graphs with given edge-density
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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 (1−1k)(n2), where k≥3 is an integer, contains at least
(110−12k+1k2−1k3+25k4)n5+o(n5)
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.
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This preprint is made available through arxiv:
https://doi.org/10.48550/arXiv.1803.00165.
Published as Bennett P, Dudek A, Lidický B, Pikhurko O. Minimizing the number of 5-cycles in graphs with given edge-density. Combinatorics, Probability and Computing. 2020;29(1):44-67. doi:10.1017/S0963548319000257