  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 $5$.
