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

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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Correspondence to John M. Sullivan.

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Izmestiev, I., Kusner, R.B., Rote, G. et al. There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems. Geom Dedicata 166, 15–29 (2013). https://doi.org/10.1007/s10711-012-9782-5

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Keywords

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

Mathematics Subject Classification (2000)

  • 05C10
  • 30F10
  • 57M50