## 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.
Last update: August 15, 2017.