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

Abstract

Sphere and projective plane triangulations a b s t r a c t 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 G a maximal null coloring f is such that the quotient graph G/f is acyclic. As an application, for maximal planar graphs (sphere triangulations) of order n ≥ 4, we prove that a vertex-coloring containing no rainbow faces uses at most  2n−1 3  colors, and this is best possible. For maximal graphs embedded on the projective plane we obtain the analogous best bound  2n+1 3  .

On the balanced upper chromatic number of cyclic projective planes and projective spaces

“…Thus the total number of lines which are not rainbow is n 2 if n 3 = 0, and at most n 2 + 3n 3 if n 1 = 0. In both cases, according to (2), we obtain less than q 2 + q + 1 not rainbow lines, contradicting that the coloring is rainbow-free.…”

“…The upper chromatic number has been studied in many different contexts and has been redefined several times under different names (see [2,3,6,7,10,13,14] and references therein). More specifically, results in projective planes appear for example in [1,4,5].…”