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Gromov hyperbolicity in lexicographic product graphs

  • Walter Carballosa
  • Amauris de la Cruz
  • José M RodríguezEmail author
Article
  • 22 Downloads

Abstract

If X is a geodesic metric space and \(x_1,x_2,x_3\in X\), a geodesic triangle \(T=\{x_1,x_2,x_3\}\) is the union of the three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) in X. The space X is \(\delta \)-hyperbolic (in the Gromov sense) if any side of T is contained in a \(\delta \)-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by \(\delta (X)\) the sharp hyperbolicity constant of X, i.e. \(\delta (X)=\inf \{\delta \ge 0: \, X \, \text { is }\delta \text {-hyperbolic}\}\). In this paper, we characterize the lexicographic product of two graphs \(G_1\circ G_2\) which are hyperbolic, in terms of \(G_1\) and \(G_2\): the lexicographic product graph \(G_1\circ G_2\) is hyperbolic if and only if \(G_1\) is hyperbolic, unless if \(G_1\) is a trivial graph (the graph with a single vertex); if \(G_1\) is trivial, then \(G_1\circ G_2\) is hyperbolic if and only if \(G_2\) is hyperbolic. In particular, we obtain the sharp inequalities \(\delta (G_1)\le \delta (G_1\circ G_2) \le \delta (G_1) + 3/2\) if \(G_1\) is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

Keywords

Lexicographic product graphs geodesics Gromov hyperbolicity infinite graphs 

Mathematics Subject Classification

Primary: 05C76 05C10 Secondary: 05C35 05C63 05C12 

Notes

Acknowledgements

The authors would like to thank the referee for his/her valuable comments which has improved the presentation of the paper considerably. This work was supported in part by two Grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain.

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Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
  2. 2.Department of MathematicsMiami Dade CollegeMiamiUSA
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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