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 ⌊ 2 n − 1 3 ⌋ 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.