A planar graph $G$ has some set of embeddings $\{E_\gamma:G \hookrightarrow R^2\}$. Each of these embeddings has associated with it a geometric dual graph $G^*_\gamma$. Using $\chi$ to denote chromatic number:

**Question 1:** For a given graph $G$, what is known about the set of integers
$\{\chi(G^*_\gamma)\}$? Is it, for instance, guaranteed to be an
interval?

A similar question can be asked about cellular embeddings of $G$ on higher-genus surfaces:

**Question 2:** For a given $G$, what is known about the chromatic numbers of the
surface duals of cellular embeddings of $G$? What about those
embeddings which share the same genus?