Facial achromatic number of triangulations on the sphere

A (proper) n-coloring c:V(G)→{1,…,n} of a graph G on a surface is a (proper) facial t-completen-coloring if every t-tuple of colors appears on the boundary of some face of G. The maximum number n such that G has a (proper) facial t-complete n-coloring is called the (proper) facial t-achromatic number. In this paper, we shall consider the (proper) facial 3-achromatic number of triangulations on the sphere. We first show that if the number of vertex disjoint faces of an even triangulation G on the sphere is at least 4n3, then G has a proper facial 3-complete n-coloring, where a triangulation is even if its each vertex has even degree. We also verify that the cubic order of the estimation is best possible and this theorem does not hold in general for 4-chromatic triangulations on the sphere. Then we completely characterize even triangulations on the sphere with proper facial 3-achromatic number exactly 3. Furthermore, we investigate the similar concept for hypergraphs.