We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3-connected plane graph with n vertices, then the number of colors in such a coloring does not exceed 7n−8
9. If G is 4-connected, then the number of colors is at most 5n−6 8 , and for n ≡ 3 (mod 8), it is at most 5n−6 8 − 1. Finally, if G is 5-connected, then the number of colors is at most 25 58 n − 22 29 . The bounds for 3-connected and 4-connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. c