Coloring vertices and faces of maps on surfaces

The vertex-face chromatic number of a map on a surface is the minimum integer m such that the vertices and faces of the map can be colored by m colors in such a way that adjacent or incident elements receive distinct colors. The vertex-face chromatic number of a surface is the maximal vertex-chromatic number for all maps on the surface. We give an upper bound on the vertex-face chromatic number of the surfaces of Euler genus ≥ 2 . The upper bound is less (by 1) than Ringel’s upper bound on the 1-chromatic number of a surface for about 5 / 12 of all surfaces. We show that there are good grounds to suppose that the upper bound on the vertex-face chromatic number is tight.