TRIAD COLORINGS OF TRIANGULATIONS ON CLOSED SURFACES

Let G be a triangulation on a closed surface. A coloring c :V(G) -> Z(n) (n >= 3) is called an n-triad coloring if for each face uvw of G, {c(u), c(v), c(w)} is equal to {i, i + 1, i + 2} for some i is an element of Z(n). We shall show that, given a natural number n >= 5, then a triangulation G on the sphere or the projective plane has an n-triad coloring if and only if G is 3-colorable. This implies that any triangulation on the sphere has either a 3- or 4-triad coloring, but there are many triangulations on the projective plane that do not have an n-triad coloring for any natural number n >= 3.