Colouring quadrangulations of projective spaces

Matěj Stehlík

Abstract

A projective quadrangulation is a graph embedded in the projective plane so that all faces are bounded by four edges. A striking theorem of Youngs asserts that every projective quadrangulation has chromatic number 2 or 4. We will extend the definition of projective quadrangulation to the n-dimensional projective space, and show that the chromatic number of an n-dimensional projective quadrangulation is either 2 or at least n+2. The proof relies on topological methods, namely the Borsuk-Ulam theorem.

We will conclude the talk by applying our result to generalised Mycielski graphs and Kneser graphs.

This is joint work with Tomáš Kaiser.