If a graph embeds in the torus or the projective plane is there an upper bound on the number of edge crossings it has in the plane?



If a graph embeds in the projective plane or the torus is there a bound on the number of edge crossings it has in the plane?
No, there is no upper bound. For any $n$, if you take a sufficiently fine mesh on a torus or projective plane, deleting $n$ edges will still result in a graph with a $K\_7$ or $K\_5$ minor, respectively.


