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Gromov hyperbolicity and convex tessellation graph

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Abstract

In [7] the following conjecture is stated: every tessellation of the Euclidean plane with convex tiles induces a non-hyperbolic graph. It is natural to think that this statement holds since the Euclidean plane is non-hyperbolic. Furthermore, there are several results supporting this conjecture. However, this work shows that the conjecture is false.

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References

  1. J. Alonso, T. Brady, D. Cooper, T. Delzant, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro and H. Short, Notes on word hyperbolic groups, in: Group Theory from a Geometrical Viewpoint (Trieste, 1990), (ed. E. Ghys, A. Haefliger and A. Verjovsky), World Sci. Publ. (River Edge, NJ, 1991), pp. 3–63.

  2. Bermudo S., Rodríguez J.M., Sigarreta J.M., Vilaire J.-M.: Gromov hyperbolic graphs. Discrete Math., 313, 1575–1585 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. B.H. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces, in: Group Theory from a Geometrical Viewpoint (Trieste, 1990), (ed. E. Ghys, A. Haefliger and A. Verjovsky), World Sci. Publ. (River Edge, NJ, 1991), pp. 64–167.

  4. Cantón A., Granados A., Pestana D., Rodríguez J.M.: Gromov hyperbolicity of periodic planar graphs. Acta Math. Sin., 30, 79–90 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cantón A., Granados A., Pestana D., Rodríguez J.M.: Gromov hyperbolicity of planar graphs. Cent. Eur. J. Math. 11, 1817–1830 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Cantón A., Granados A., Pestana D., Rodríguez J.M.: Gromov hyperbolicity of periodic graphs. Bull. Malays. Math. Sci. Soc., 39, 89–116 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carballosa W., Portilla A., Rodríguez J.M. Sigarreta J.M.: Planarity and hyperbolicity in graphs. Graph Combinator., 31, 1311–1324 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Chepoi and B. Estellon, Packing and covering δ-hyperbolic spaces by balls, APPROX-RANDOM 2007, pp. 59–73.

  9. D. Eppstein, Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition, in: SODA’2007, Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms, ACM (New York, 2007), pp. 29-38.

  10. E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser, Boston Inc. (Boston, MA, 1990).

  11. M. Gromov, Hyperbolic groups, in: Essays in Group Theory, Edited by S. M. Gersten, M. S. R. I. Publ. 8, Springer (New York, 1987), pp. 75–263.

  12. E. A. Jonckheere, Contrôle du traffic sur les réseaux à géométrie hyperbolique–Vers une théorie géométrique de la sécurité l’acheminement de l’information, J. Europ. Syst. Autom., 8 (2002), 45–60.

  13. D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat and M. Boguñá, Hyperbolic geometry of complex networks, Physical Review E, 82 (2010), 036106, 18 pp.

  14. K. Oshika, Discrete Groups, AMS Bookstore (2002).

  15. A. Portilla, J. M. Rodríguez, J. M. Sigarreta and J.-M. Vilaire, Gromov hyperbolic tessellation graphs, Utilitas Math. (to appear).

  16. Y. Shavitt and T. Tankel, On internet embedding in hyperbolic spaces for overlay construction and distance estimation, INFOCOM (2004).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, Miami, FL, U.S.A.

    W. Carballosa

  2. Department of Mathematics, Miami Dade College, Miami, FL, U.S.A.

    W. Carballosa

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  1. W. Carballosa
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Correspondence to W. Carballosa.

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Carballosa, W. Gromov hyperbolicity and convex tessellation graph. Acta Math. Hungar. 151, 24–34 (2017). https://doi.org/10.1007/s10474-016-0677-z

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  • DOI: https://doi.org/10.1007/s10474-016-0677-z

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