Planar Turán Numbers of Cycles: A Counterexample

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2021-10-05
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Cranston, Daniel
Lidicky, Bernard
Liu, Xiaonan
Shantanam, Abhinav
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© Author(s) 2021
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Mathematics
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Mathematics
Abstract
The planar Turan number exP(Cℓ,n) is the largest number of edges in an n-vertex planar graph with no ℓ-cycle. For ℓ∈{3,4,5,6}, upper bounds on exP(Cℓ,n) are known that hold with equality infinitely often. Ghosh, Györi, Martin, Paulo, and Xiao [arXiv:2004.14094] conjectured an upper bound on exP(Cℓ,n) for every ℓ≥7 and n sufficiently large. We disprove this conjecture for every ℓ≥11. We also propose two revised versions of the conjecture.
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This preprint is made available through arXiv:https://arxiv.org/abs/2110.02043. This work is licensed under the Creative Commons Attribution 4.0 License.
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