The Crossing Number of a graph is the minimum value of the crossing point among all drawings...
on the other hand, using Euler's formula, we know that a graph is embeddable in a space with a sufficiently large genus.
Since we can consider each hole in (high genus) space as a bridge (handle) that some edges can go through, we know that $cr(G) \geqslant g(G)$. But is there any better inequality between them?
Thanks in advance.
As you have observed, for any graph $G$, we always have $$cr(G)\geq g(G)\tag{1}$$ where $cr(G)$ is the crossing number of $G$ and $g(G)$ is the genus of $G$.
On the other hand, the difference between the crossing number and the genus of a graph can be arbitrarily large, as we can see from this example: The crossing number of $C_3\times C_n$, the Cartesian product of the cycle $C_3$ and $C_n$, is given by $cr(C_3\times C_n)=n$ for $n\geq 3$ (see here) . On the other hand, it is easy to show that the genus of $C_3\times C_n$ is given by $g(C_3\times C_n)=1$ for $n\geq 3$.
So I would say $(1)$ is the only relationship we have between crossing number and genus.
