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There is no 5,7-triangulation of the torus, that is, no triangulation withexactly two exceptional vertices, of degree 5 and 7. Similarly, there is no3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot bebicolored. Similar statements hold for 4,8-triangulations and2,6-quadrangulations. We prove these results, of which the first two are knownand the others seem to be new, as corollaries of a theorem on the holonomygroup of a euclidean cone metric on the torus with just two cone points. Weprovide two proofs of this theorem: One argument is metric in nature, the otherrelies on the induced conformal structure and proceeds by invoking the residuetheorem. Similar methods can be used to prove a theorem of Dress on infinitetriangulations of the plane with exactly two irregular vertices. Thenon-existence results for torus decompositions provide infinite families ofgraphs which cannot be embedded in the torus.

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems