Grunbaum colorings of even triangulations on surfaces

A Grunbaum coloring of a triangulation G is a map c : E(G){1,2,3} such that for each face f of G, the three edges of the boundary walk of f are colored by three distinct colors. By Four Color Theorem, it is known that every triangulation on the sphere has a Grunbaum coloring. So, in this article, we investigate the question whether each even (i.e.,Eulerian) triangulation on a surface with representativity at least r has a Grunbaum coloring. We prove that, regardless of the representativity, every even triangulation on a surface F has a Grunbaum coloring as long as F is the projective plane, the torus, or the Klein bottle, and we observe that the same holds for any surface with sufficiently large representativity. On the other hand, we construct even triangulations with no Grunbaum coloring and representativity r=1,2, and 3 for all but finitely many surfaces. In dual terms, our results imply that no snark admits an even map on the projective plane, the torus, or the Klein bottle, and that all but finitely many surfaces admit an even map of a snark with representativity at least3