Counting k-cycles in $5$-connected planar triangulations

We show that every n-vertex $5$-connected planar triangulation has at most $9n-50$ many cycles of length $5$ for all n≥ 20 and this upper bound is tight. We also show that for every k≥ 6, there exists some constant $C(k)$ such that for sufficiently large n, every n-vertex $5$-connected planar graph has at most C(k) · n^k/3 many cycles of length k. This upper bound is asymptotically tight for all k≥ 6.