  If all but two vertices of a triangulated sphere have degrees divisible by
$k$, then the exceptional vertices are not adjacent. This theorem is proved for
$k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by
the vertices of platonic solids have to be used. With a coloring monodromy one
can associate a branched cover. This generalizes to a space of germs between
two triangulated surfaces. We also discuss relations with Belyi surfaces and
with cone-metrics of constant curvature.
