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    $\begingroup$ I guess that you mean that a (cartographic) map on a Klein bottle is colorable by 6 colors, etc. This is mentioned (and generalized) in the Wikipedia article: en.wikipedia.org/wiki/Four_color_theorem#Generalizations $\endgroup$ Commented Dec 8, 2014 at 3:36
  • $\begingroup$ Indeed so. How about if each country may consist of n disjoint regions? $\endgroup$ Commented Dec 8, 2014 at 8:51
  • $\begingroup$ $K_{7}$ in the torus has a nice algebraic description: Start with the graph formed by the Eisenstein integers, where $a$ is adjacent to $b$ means $|a-b| = 1$. Every vertex of this graph has degree 6, and it's planar. To make the plane a torus, we quotient by a lattice. To make this well-defined for how we identify vertices and edges, that lattice has to be an ideal. Choose an ideal of norm 7, like $(2+\sqrt{-3})$, and we now have 7 vertices (all have degree 6), yielding $K_{7}$. I don't know if replacing the Eisenstein ring with structures like the Hurwitz ring gives nice generalizations. $\endgroup$ Commented Dec 9, 2014 at 20:14
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    $\begingroup$ The problem is only interesting for surfaces because every graph can be embedded in a manifold of dimension 3 or higher. $\endgroup$ Commented Dec 18, 2014 at 20:46