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 problem about whether there is a similar result for planar bipartite graphs. For example, an **upper bound** on the number of **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.

 - [The upper bound on the number of four cycles of 3-connected quadrangulation graphs.][1]


  [1]: https://math.stackexchange.com/questions/4423841/the-upper-bound-on-the-number-of-four-cycles-of-3-connected-quadrangulation-grap