# Clique vs genus

Is there a relation between Clique size $\omega(G)$ and genus $g(G)$ of a graph? That is does $$\omega(G)^c\geq g(G)\geq \omega(G)^{\frac{1}d}$$ hold with constants $c,d\geq1$?

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The genus of the complete graph on $n \geq 3$ vertices $K_{n}$ is:
$$\gamma(n) = \left\lceil\frac{(n-3)(n-4)}{12}\right\rceil$$
We can subdivide the edges of $K_{n}$ to obtain a graph $H$ with clique-number $\omega(H) = 2$ (or anything from $2$ to $n$ as we choose), but of course this still has a $K_{n}$ minor, and hence the same genus as $K_{n}$.