  There is no 5,7-triangulation of the torus, that is, no triangulation with
exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no
3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be
bicolored. Similar statements hold for 4,8-triangulations and
2,6-quadrangulations. We prove these results, of which the first two are known
and the others seem to be new, as corollaries of a theorem on the holonomy
group of a euclidean cone metric on the torus with just two cone points. We
provide two proofs of this theorem: One argument is metric in nature, the other
relies on the induced conformal structure and proceeds by invoking the residue
theorem. Similar methods can be used to prove a theorem of Dress on infinite
triangulations of the plane with exactly two irregular vertices. The
non-existence results for torus decompositions provide infinite families of
graphs which cannot be embedded in the torus.

Combinatorics;Geometric Topology