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

TL;DR

This paper investigates the maximum number of colors in vertex colorings of planar and projective plane graphs that avoid rainbow faces, introducing null colorings and using homological tools to establish optimal bounds.

Contribution

It introduces the concept of null colorings for graphs and proves bounds on the number of colors in rainbow-free colorings of planar and projective plane graphs.

Findings

Maximum colors in rainbow-free colorings of planar graphs: 

Maximum colors in rainbow-free colorings of projective plane graphs: 

Introduction of null colorings and their properties in topological graph theory.

Abstract

For maximal planar graphs of order $n\geq 4$, we prove that a vertex--coloring containing no rainbow faces uses at most $\lfloor\frac{2n-1}{3}\rfloor$ colors, and this is best possible. For maximal graph embedded on the projective plane, we obtain the analogous best bound $\lfloor\frac{2n+1}{3}\rfloor$. The main ingredients in the proofs 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.