Suppose a delaunay triangulation over the surface of a 3d sphere (or generally some 3d surface of something topologically equivalent to the sphere). How many colors do I need to color its triangles so that triangles sharing an edge have different colors?
My idea: For the delaunay triangulation of a set of points on the plane, 3 colors are always enough.
Proof: taking the 1-ring (all the triangles that touch an epsilon small circle around a vertex) 3 colors are always enough to color it.
This is taken from wikipedia. The 4rth point from the bottom, (4rth in the sense of y coordinate) has a 1-ring of size 5, thus I need three colors to color it.
I think the argument still holds for the surface of a sphere. Am I correct?
Will it still hold even if there is genus?