Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on a Fixed Surface

Thomassen, Carsten

doi:10.2307/2001547

Cited by 45 publications

(55 citation statements)

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“…This observation allows us to give an easy proof of the classification of (degree-)regular tilings of the torus. This has been treated by many authors [2,16,17,5,4], although most of the results are restricted to the class of polyhedral maps.…”

Section: Euclidean Cone Metricsmentioning

confidence: 99%

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

et al. 2012

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There is no triangulation of the torus with vertex degrees 5, 6, . . . , 6, 7 and related results: Geometric proofs for combinatorial theoremsIvan Izmestiev · Robert B. Kusner · Günter Rote · Boris Springborn · John M. Sullivan 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. IntroductionIn any triangulation of the torus, the average vertex degree is 6, so vertices of degree d = 6 can be considered exceptional. It is easy to find regular triangulations with no exceptional vertices, as in Figure 1. Applying a single edge flip to such a triangulation produces a triangulation with four exceptional vertices (assuming the four vertices in question are distinct): two of degree 5 and two of degree 7, as in Figure 2(a). We call this a 5 2 7 2 -triangulation. Similarly, we can produce examples of triangulations with just two exceptional vertices, assuming these have degrees other than 5 and 7. Figure 3 shows 4,8-, 3,9-, 2,10-and 1,11-triangulations of the torus. However:The torus has no 5,7-triangulation, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7.We can also consider quadrangulations of the torus. In this case, the average vertex degree is 4, and an analogous theorem holds:Theorem 2 (Barnette, Jucovič & Trenkler [3]) The torus has no 3,5-quadrangulation, that is, no quadrangulation with exactly two exceptional vertices, of degree 3 and 5.On the other hand, 2,4-and 3 2 5 2 -quadrangulations do exist, as shown in Figure 4. Finally, one

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Section: Euclidean Cone Metricsmentioning

confidence: 99%

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

et al. 2012

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“…Regular and chiral maps have been considered from the viewpoints of the surfaces (see for example [6], [11] and [13]), of the automorphism groups (see [46]) and of the graphs (see [7], [28] and [62]). However little has been said about which regular and chiral maps are polyhedra.…”

Section: Rankmentioning

confidence: 99%

Developments and open problems on chiral polytopes

Pellicer¹

2012

Ars Math. Contemp.

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“…For example, Thomassen's conjecture [15,122] says that only finitely many connected vertex-transitive graphs without a Hamilton cycle exist, whereas Babai's conjecture [12,13] says that infinitely many such graphs exist. A large number of articles directly or indirectly related to this problem (for the list of relevant references and a detailed description of the status of this problem see [67]), have appeared in the literature, affirming the existence of such paths in some special vertex-transitive graph and, in some cases, also the existence of Hamilton cycles.…”

Section: Hamilton Paths and Cyclesmentioning

confidence: 99%

Recent trends and future directions in vertex-transitive graphs

Kutnar

Marušič

2008

Ars Math. Contemp.

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A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade.Keywords: Vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group.Math. Subj. Class.: 05C25, 20B25 1 In the beginning Vertex-transitive graphs, that is, graphs whose automorphisms groups acts transitively on the corresponding vertex sets, have been an active topic of research for a long time now. Much of this interest is due to their suitability to model scientific phenomena when symmetry is an issue. It is the aim of this article to discuss recent developments surrounding two well known open problems in vertex-transitive graphs -the Hamilton path/cycle problem and the semiregularity problem -as well as the general question addressing structural properties of arc-transitive and half-arc-transitive graphs. In doing so we will also try to contemplate possible directions this area of research is likely to take in the near future. * Supported by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285.E-mail addresses: klavdija.kutnar@upr.si (Klavdija Kutnar), dragan.marusic@upr.si (Dragan Marušič) Copyright c 2008 DMFA -založništvo K. Kutnar, D. Marušič: Recent Trends and Future Directions in Vertex-Transitive Graphs 113The article is organized as follows. In Section 2 we discuss the problem, posed by the second author in 1981 (see [79]) who asked if it is true that a vertex-transitive digraph contains a nonidentity automorphism with all orbits of equal length, in short, a semiregular automorphism. Section 3 gives a quick overview of the problem, posed by Lovàsz in 1969 (see [72]), who asked if it is true that every connected vertex-transitive graph contains a Hamilton path, thus motivating a great deal of research into vertex-transitive graphs in the following decades. In Section 4, imprimitivity, one of the most fundamental concepts in the theory of permutation groups, is considered with special emphasis given to a recently developed theory which links the existence of blocks of imprimitivity in vertex-transitive graphs, having an abelian semiregular subgroup of automorphisms, to certain conditions that need to be satisfied in the corresponding quotient graph relative to the orbits of such a subgroup. Finally, in Section 5 we deal with some recent structural results on arc-transitive and half-arc-transitive graphs, as well as their link to some open problems in the theory of configurations.For group-theoretic terms not defined here we refer the reader to [124]. Semiregularit...

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Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on a Fixed Surface

Cited by 45 publications

References 4 publications

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 triangulation of the torus with vertex degrees 5, 6, ... , 6, 7 and related results: geometric proofs for combinatorial theorems

Developments and open problems on chiral polytopes

Recent trends and future directions in vertex-transitive graphs

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