There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems
| Autor: | Izmestiev, Ivan, Kusner, Robert B., Rote, Günter, Springborn, Boris, Sullivan, John M. |
|---|---|
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Geom. Dedicata 166:1 (2013), 15-29 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s10711-012-9782-5 |
| Popis: | 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. Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicata |
| Databáze: | arXiv |
| Externí odkaz: |
|