In 1979, Hakimi and Schmeichel [1] initiated such a study by determining the maximum number of triangles and 4-cycles possible in an $n$-vertex planar graph (see also [2] for a small correction).
[1] S. Hakimi, E. Schmeichel. On the Number of Cycles of Length k in a Maximal Planar Graph. J. Graph Theory 3 (1979): 69–86.
[2] A. Alameddine. On the Number of Cycles of Length 4 in a Maximal Planar Graph. J. GraphTheory, 4(1980): 417–422.
In 2019, Győri et al. [3] studied the maximum number of pentagons on planar graphs.
- [3] Győri E, Paulos A, Salia N, et al. The maximum number of pentagons in a planar graph[J]. arXiv preprint arXiv:1909.13532, 2019.
I have a probelem about whether there is a similar result for planar bipartite graphs. For example, an upper bound on 4 cycles (or any other even cycles). I haven't found it yet. If not, it is not clear what the difficulty is.
I once asked a similar question at Math. Stack and did not seem to see any results.