The planar Turán number exP(C_,n) is the largest number of edges in an n-vertex planar graph with no -cycle. For each ∈ \3,4,5,6\, upper bounds on exP(C_,n) are known that hold with equality infinitely often. Ghosh, Győri, Martin, Paulos, and Xiao [arXiv:2004.14094] conjectured an upper bound on exP(C_,n) for every ≥ 7 and n sufficiently large. We disprove this conjecture for every ≥ 11. We also propose two revised versions of the conjecture.
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Cranston et al. (2022) studied this question.