In topology, being able to decompose manifolds into simplicial complexes is of great importance to better study their properties. As such, it is vital that techniques be developed to compute topological invariants between different manifolds. This project will attempt to examine a property known as Grunbaum coloring, and determine when one can find such a coloring given a triangulation of a manifold. It is known when Grunbaum coloring exist for the torus, sphere, and projective plane, but more work can be done on 3-manifolds or surfaces of genus 4 and greater. The results from this research may have implications in graph theory or computations involving triangulations, as homeomorphisms are more easily computable.