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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References
Ahlfors L.V.: Complex Analysis. 3rd edn. McGraw-Hill Book Co., New York (1978)
Altshuler A.: Construction and enumeration of regular maps on the torus. Discret. Math. 4, 201–217 (1973)
Barnette D., Jucovič E., Trenkler M.: Toroidal maps with prescribed types of vertices and faces. Mathematika 18, 82–90 (1971)
Brehm U., Kühnel W.: Equivelar maps on the torus. Eur. J. Combin. 29(8), 1843–1861 (2008). doi:10.1016/j.ejc.2008.01.010
Datta B., Upadhyay A.K.: Degree-regular triangulations of torus and Klein bottle. Proc. Indian Acad. Sci. Math. Sci 115(3), 279–307 (2005)
Bobenko, A.I., Kenyon, R.W., Sullivan, J.M., Ziegler, G.M. (eds.): Discrete differential geometry. Oberwolfach Rep. 3(1), 653–728 (2006)
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)
Eberhard V.: Zur Morphologie der Polyeder. Teubner, Leipzig (1891)
Gagarin A., Myrvold W., Chambers J.: The obstructions for toroidal graphs with no K 3,3’s. Discret. Math. 309(11), 3625–3631 (2009). doi:10.1016/j.disc.2007.12.075
Grünbaum B.: Some analogues of Eberhard’s theorem on convex polytopes. Israel J. Math. 6, 398–411 (1968)
Grünbaum B.: Planar maps with prescribed types of vertices and faces. Mathematika 16, 28–36 (1969)
Grünbaum, B.: Convex polytopes, Graduate Texts in Mathematics, vol. 221. Springer, New York. 2nd edn. prepared by Kaibel, Klee and Ziegler (2003)
Grünbaum B., Szilassi L.: Geometric realizations of special toroidal complexes. Contrib. Discret. Math. 4(1), 21–39 (2009)
Jendrol’ S., Jucovič E.: On the toroidal analogue of Eberhard’s theorem. Proc. Lond. Math. Soc. (3) 25, 385–398 (1972)
Jucovič E., Trenkler M.: A theorem on the structure of cell-decompositions of orientable 2-manifolds. Mathematika 20, 63–82 (1973)
Negami S.: Uniqueness and faithfulness of embedding of toroidal graphs. Discret. Math. 44(2), 161–180 (1983)
Thomassen C.: Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface. Trans. Am. Math. Soc 323(2), 605–635 (1991)
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)
Zaks J.: The analogue of Eberhard’s theorem for 4-valent graphs on the torus. Israel J. Math. 9, 299–305 (1971)
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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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DOI: https://doi.org/10.1007/s10711-012-9782-5
Keywords
- Torus triangulation
- Euclidean cone metric
- Holonomy
- Meromorphic differential
- Residue theorem
- Burgers vector