Cyclic 4-Colorings of Graphs on Surfaces

To attack the Four Color Problem, in 1880, Tait gave a necessary and sufficient condition for plane triangulations to have a proper 4-vertex-coloring: a plane triangulation G has a proper 4-vertex-coloring if and only if the dual of G has a proper 3-edge-coloring. A cyclic coloring of a map G on a surface F 2 is a vertex-coloring of G such that any two vertices x and y receive different colors if x and y are incident with a common face of G. In this paper, we extend the result by Tait to two directions, that is, considering maps on a non-spherical surface and cyclic 4-colorings.