 A Fisk graph: a geometric surface with chromatic number 5. The embedding story suggests that 5 is the upper limit for all surfaces. (Like one can embed a one dimensional closed curve into the boundary
of a three dimensional ball so that the complement in the open ball remains simply connected, it is possible to write
an orientable surface into the three sphere bounding a four dimensional ball in such a way that the open inside of the later
remains simply connected. In the nonorientable case, the embedding has to leave the 3 sphere at some places but we can
keep the complement in the open four dimensional ball simply connected).
To construct an example of the torus with chromatic number 5, start with a flat triangulated torus
and cut rhomboids to get a geodesic with winding number (2,1).
Now cut the torus along a circle and Dehn twist it so that the geodesic is broken.
Now two adjacent vertices have odd degrees. It was Fisk who noticed that this leads to chromatic number 5.

The 16 cell is the smallest three dimensional sphere. It is one of the 6 platonic solids in 4 dimensions, the analogue of the octahedron.
Its dual is the tesseract. It can be colored with 4 colors which is the minimal number of colors for three dimensional spheres. We believe that any 3sphere can be colored with 4 or 5 colors. 
