Ivan Izmestiev, Robert B. Kusner, Günter Rote, Boris Springborn, and John M. Sullivan:

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

Geometriae Dedicata 166 (2013), 15–29. doi:10.1007/s10711-012-9782-5, arXiv:1207.3605 [math.CO].

Abstract

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.

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