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For maximal planar graphs of order $n\geq 4$, we prove that avertex--coloring containing no rainbow faces uses at most$\lfloor\frac{2n-1}{3}\rfloor$ colors, and this is best possible. For maximalgraph embedded on the projective plane, we obtain the analogous best bound$\lfloor\frac{2n+1}{3}\rfloor$. The main ingredients in the proofs areclassical homological tools. By considering graphs as topological spaces, weintroduce the notion of a null coloring, and prove that for any graph $G$ amaximal null coloring $f$ is such that the quotient graph $G/f$ is a forest.

Null and non--rainbow colorings of projective plane and sphere triangulations