In 5-connected planar triangulations, the maximum number of 5-cycles is 9n-50, and for k≥6 the maximum number of k-cycles is Θ(n^{⌊k/3⌋}).
Borodin, On the total coloring of planar graphs, J
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Counting $k$-cycles in $5$-connected planar triangulations
In 5-connected planar triangulations, the maximum number of 5-cycles is 9n-50, and for k≥6 the maximum number of k-cycles is Θ(n^{⌊k/3⌋}).