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

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2018-03-01
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Bennett, Patrick
Dudek, Andrzej
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Lidicky, Bernard
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Mathematics
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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 cycles 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 for 2≤k≤73.

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This is a manuscript made available through arxiv: https://arxiv.org/abs/1803.00165.

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Mon Jan 01 00:00:00 UTC 2018
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