Non-rainbow colorings of 3-, 4- and 5-connected plane graphs

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 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 ⌊ 43 100n − 19 25 ⌋ . 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.

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