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

By considering graphs as topological spaces we introduce, at the level of homology, the notion of a null coloring, which provides new information on the task of clarifying the structure of cycles in a graph. We prove that for any graph GG a maximal null coloring ff is such that the quotient graph G/fG/f is acyclic. As an application, for maximal planar graphs (sphere triangulations) of order n≥4n≥4, we prove that a vertex-coloring containing no rainbow faces uses at most ⌊2n−13⌋ colors, and this is best possible. For maximal graphs embedded on the projective plane we obtain the analogous best bound ⌊2n+13⌋.