Geometriae Dedicata

, Volume 166, Issue 1, pp 15–29 | Cite as

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

  • Ivan Izmestiev
  • Robert B. Kusner
  • Günter Rote
  • Boris Springborn
  • John M. Sullivan
Original Paper


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.


Torus triangulation Euclidean cone metric Holonomy Meromorphic differential Residue theorem Burgers vector 

Mathematics Subject Classification (2000)

05C10 30F10 57M50 


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Ivan Izmestiev
    • 1
  • Robert B. Kusner
    • 2
  • Günter Rote
    • 3
  • Boris Springborn
    • 4
  • John M. Sullivan
    • 5
  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.University of MassachusettsAmherstUSA
  3. 3.Freie Universität BerlinBerlinGermany
  4. 4.Technische Universität MünchenGarchingGermany
  5. 5.Technische Universität BerlinBerlinGermany

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