There is no triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems
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.
KeywordsTorus triangulation Euclidean cone metric Holonomy Meromorphic differential Residue theorem Burgers vector
Mathematics Subject Classification (2000)05C10 30F10 57M50
Unable to display preview. Download preview PDF.
- 1.Ahlfors L.V.: Complex Analysis. 3rd edn. McGraw-Hill Book Co., New York (1978)Google Scholar
- 6.Bobenko, A.I., Kenyon, R.W., Sullivan, J.M., Ziegler, G.M. (eds.): Discrete differential geometry. Oberwolfach Rep. 3(1), 653–728 (2006)Google Scholar
- 7.Dress, A.W.M.: On the classification of local disorder in globally regular spatial patterns. In: Temporal order, Springer Ser. Synergetics, vol. 29, pp. 61–66. Springer, Berlin (1985)Google Scholar
- 8.Eberhard V.: Zur Morphologie der Polyeder. Teubner, Leipzig (1891)Google Scholar
- 12.Grünbaum, B.: Convex polytopes, Graduate Texts in Mathematics, vol. 221. Springer, New York. 2nd edn. prepared by Kaibel, Klee and Ziegler (2003)Google Scholar
- 18.Thurston, W.P.: Shapes of polyhedra and triangulations of the sphere. In: The Epstein Birthday Schrift, Geom. Topol. Monogr., vol. 1, pp. 511–549. Geom. Topol. Publ., Coventry (1998)Google Scholar