Null and non–rainbow colorings of maximal planar graphs

For maximal planar graphs of order n⩾4, we prove that a vertex–coloring containing no rainbow faces uses at most colors, and this is best possible. The main ingredients in the proof are classical homological tools. By considering graphs as topological spaces, we introduce the notion of a null coloring, and prove that for any graph G a maximal null coloring f is such that the quotient graph G/f is a forest.