Existence of a small cover over a 15-colorable simple 4-polytope

Abstract

The chromatic number for properly colouring the facets of a combinatorial simple $n$-polytope $P^n$ that is the orbit space of a quasitoric manifold satisfies the inequality $n\leq P^n\leq 2^n-1$. The inequality is sharp for $n=2$ but not for $n=3$ due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals $15$ and deduce that the predicted upper bound is attained for $n=4$. Analogues results are verified for the case of oriented small covers in dimensions $4$ and $5$.