Color or cover

Abstract

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.