Planar Turan Numbers of Cycles: A Counterexample

The planar Turan number exP(C-l, n) is the largest number of edges in an nvertex planar graph with no l-cycle. For each l is an element of{3; 4; 5; 6}, upper bounds on exP(C-l; n) are known that hold with equality infinitely often. Ghosh, Gyori, Martin, Paulos, and Xiao [arXiv:2004.14094] conjectured an upper bound on exP(C-l,C- 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.

Automated summary

This paper discusses the problem of planar graphs without certain length cycles, providing upper bounds on the number of edges for cycles of length 3, 4, 5, and 6, and proposing a conjecture for cycles of length greater than or equal to 7. We disprove this conjecture for cycles of length greater than or equal to 11 and propose revised versions of the conjecture.