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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 11, Issue 10

Issues

Volume 13 (2015)

Gromov hyperbolicity of planar graphs

Alicia Cantón / Ana Granados / Domingo Pestana
  • Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain
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/ José Rodríguez
  • Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain
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Published Online: 2013-07-20 | DOI: https://doi.org/10.2478/s11533-013-0286-9

Abstract

We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ2 with tiles which are parallelograms would be non-hyperbolic.

MSC: 05C10; 05C63; 05C75; 05A20

Keywords: Planar Graphs; Gromov Hyperbolicity; Infinite Graphs; Geodesics; Tessellation

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About the article

Published Online: 2013-07-20

Published in Print: 2013-10-01


Citation Information: Open Mathematics, Volume 11, Issue 10, Pages 1817–1830, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0286-9.

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