Irreducible triangulations of the torus

Лавренченко, С А

doi:10.1007/bf01104169

Cited by 50 publications

(45 citation statements)

References 4 publications

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“…Hereafter we will use the symbols TPn and TAn to refer to toroidal lattices with 2n disclination pairs and symmetry group D nh and D nd respectively. A systematic construction of defected triangulations of the torus can be achieved in the context of planar graphs [188,189,190]. A topological embedding of a graph in a two-dimensional manifold corresponds to a triangulation of the manifold if each region of the graph is bounded by exactly three vertices and three edges, and any two regions have either one common vertex or one common edge or no common elements of the graph.…”

Section: Geometry Of Toroidal Polyhedramentioning

confidence: 99%

Two-dimensional matter: order, curvature and defects

Bowick

Giomi

2009

Advances in Physics

340

458

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Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its Gaussian curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics controls the detailed structured of ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the form of grain boundary scars for spherical crystals above a minimal system size, dislocation unbinding for toroidal hexatics, interstitial fractionalization in spherical crystals and the appearance of well-separated disclinations for toroidal crystals. Much of the analysis leads to universal predictions that do not depend on the details of the microscopic interactions that lead to order in the first place. These predictions are subject to test by the many experimental soft and hard matter systems that lead to curved ordered structures such as colloidal particles self-assembling on droplets of one liquid in a second liquid. The defects themselves may be functionalized to create ligands with directional bonding. Thus nano to meso scale superatoms may be designed with specific valency for use in building supermolecules and novel bulk materials. Parameters such as particle number, geometrical aspect ratios and anisotropy of elastic moduli permit the tuning of the precise architecture of the superatoms and associated supermolecules. Thus the field has tremendous potential from both a fundamental and materials science/supramolecular chemistry viewpoint. Contents

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Section: Geometry Of Toroidal Polyhedramentioning

confidence: 99%

Two-dimensional matter: order, curvature and defects

Bowick

Giomi

2009

Advances in Physics

340

458

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“…Lawrencenko [14] has determined the complete set I of 21 irreducible toroidal triangulations, and the graphs K 7 , K 8 − 4K 2 , K 9 − C 9 , and K 9 − 3K 3 are exactly the 6-regular graphs in I. So already Lawrencenko's result implies that these four graphs are minor-minimal 6-regular toroidal graphs.…”

Section: Old and New Resultsmentioning

confidence: 92%

Contractions of 6-connected toroidal graphs

Fijavž

2007

Journal of Combinatorial Theory, Series B

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Let T 6 denote the class of all 6-connected (equivalently 6-regular) toroidal graphs and let G ∈ T 6 which is not minor-minimal in T 6 . Let G ∈ T 6 be a proper minor of G with maximum number of vertices. We show that |V (G)| − |V (G )| = fw(G), where fw(G) denotes the face-width of the toroidal embedding of G. Consequently, we show that the only minor-minimal graphs in T 6 are K 7 , K 8 − 4K 2 , K 9 − C 9 , and K 9 − 3K 3 .

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“…A triangulation G is irreducible if G has no contractible edge. It is known that every surface admits finitely many irreducible triangulations, up to homeomorphism , and the complete lists of irreducible triangulations are known for

S_{0}

S_{1}

S_{2}

N_{1}

N_{2}

N_{3}

, and

N_{4}

. (See for more details about irreducible triangulations.…”

Section: Applicationmentioning

confidence: 99%

Even Embeddings of the Complete Graphs and Their Cycle Parities

Noguchi

2016

Journal of Graph Theory

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The complete graph Kn on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F2 with Euler characteristic εfalse(F2false)=nfalse(5−nfalse)/4 if and only if n≡0,5(mod8) (resp. n≡0,1(mod4) and n≠5). In this article, we shall show that if Kn quadrangulates a closed surface F2, then Kn has a quadrangular embedding on F2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism ρ:π1false(F2false)→double-struckZ2, called the cycle parity.

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Irreducible triangulations of the torus

Cited by 50 publications

References 4 publications

Two-dimensional matter: order, curvature and defects

Two-dimensional matter: order, curvature and defects

Contractions of 6-connected toroidal graphs

Even Embeddings of the Complete Graphs and Their Cycle Parities

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