Matchings and Nonrainbow Colorings

We show that the maximum number of colors that can be used in a vertex coloring of a cubic 3-connected plane graph $G$ that avoids a face with vertices of mutually distinct colors (a rainbow face) is equal to $\frac{n}{2}+\mu^*-2$, where $n$ is the number of vertices of $G$ and $\mu^*$ is the size of the maximum matching of the dual graph $G^*$.