Coloring zonotopal quadrangulations of the projective space

Abstract

A quadrangulation on a surface $F^2$ is a map of a simple graph on $F^2$ such that each 2-dimensional face is quadrangular. Youngs proved that every quadrangulation on the projective plane $P^2$ is either bipartite or 4-chromatic. It is a surprising result since every quadrangulation on an orientable surface with sufficiently high edge-width is 3-colorable. Kaiser and Stehlík defined a $d$-dimensional quadrangulation on the $d$-dimensional projective space $P^d$ for any $d\ge 2$, and proved that any such quadrangulation has chromatic number at least $d+2$ if it is not bipartite. In this paper, we define another kind of $d$-dimensional quadrangulations of $P^d$ for any $d\ge 2$, and prove that such a quadrangulation $Q$ is always 4-chromatic if $Q$ is non-bipartite and satisfies a special geometric condition related to a zonotopal tiling of a $d$-dimensional zonotope.