Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G   which embeds as a quadrangulation in the real projective space PnPn has chromatic number n+2n+2 or higher, unless G   is bipartite. For n=2n=2 this was proved by Youngs (1996) [20]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovász–Kneser theorem.