$k$-planar graphs and genus

Question

Is there a simple function that connects $k$ in $k$-planar graphs and genus of such graphs?

If there is no simple function is there any non-trivial upper and lower bound?

Answer 1

For fixed $k$ (even $k=1$) the genus can be arbitrarily large (linear in the number of vertices). Graphs with genus $g$ have $3n+O(g)$ edges by Euler's formula, while 1-planar graphs can have as many as $4n-O(1)$ edges. So dense 1-planar graphs (e.g. take a planar graph with all faces quadrilaterals and add two crossing edges in each face) have high genus.

In the other direction, even the graphs of bounded genus can have unbounded local crossing number (again linear in the number of vertices). For instance, take the nonplanar (but toroidal) graph $K_{3,3}$ and replace each edge by a collection of $n$ two-edge paths ($K_{2,n}$). Then the resulting toroidal graph has quadratic crossing number (it has $\Omega(n^9)$ subdivided copies of $K_{3,3}$, each with at least one crossing, and each crossing belongs to only $O(n^7)$ subdivided copies of $K_{3,3}$) so some edge must be crossed linearly many times.

So there is no functional relationship in either direction.