Consider the set of planar graphs where all the internal faces are triangles. If there is an interior point of odd degree the graph cannot be three colored. If every interior point has even degree can it always be three colored? Ideally I'd like a small counterexample.
Yes, this is a corollary of the Three Color Theorem, see at the bottom here: http://kahuna.merrimack.edu/~thull/combgeom/colornotes.html
This result extends to high dimensions. A triangulation of a d-dimensional sphere so that every vertex has an even degree is (d+1) colorable. See, for example this paper: Jacob E. Goodman and Hironori Onishi, Even triangulations of $S^3$ and the coloring of graphs, Trans. Am. Math. Soc. 246 (1978), 501–510.