Gromov Hyperbolicity in the Cartesian Sum of Graphs

Abstract

In this paper, we characterize the hyperbolic product graphs for the Cartesian sum \(G_1\oplus G_2\): \(G_1\oplus G_2\) is always hyperbolic, unless either \(G_1\) or \(G_2\) is the trivial graph (the graph with a single vertex); if \(G_1\) or \(G_2\) is the trivial graph, then \(G_1\oplus G_2\) is hyperbolic if and only if \(G_2\) or \(G_1\) is hyperbolic, respectively. Besides, we characterize the Cartesian sums with hyperbolicity constant \(\delta (G_1\oplus G_2) = t\) for every value of t. Furthermore, we obtain the sharp inequalities \(1\le \delta (G_1\oplus G_2)\le 3/2\) for every non-trivial graphs \(G_1,G_2\). In addition, we obtain simple formulas for the hyperbolicity constant of the Cartesian sum of many graphs. Finally, we prove the inequalities \(3/2\le \delta (\overline{G_1\oplus G_2})\le 2\) for the complement graph of \(G_1\oplus G_2\) for every \(G_1,G_2\) with \(\min \{{{\mathrm{diam}}}V(G_1), {{\mathrm{diam}}}V(G_2)\}\ge 3\).

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Acknowledgements

We would like to thank the referee for his/her careful reading of this manuscript and some valuable comments which have improved the presentation of the paper. This work was supported in part by three grants from Ministerio de Economía y Competitividad (MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

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Correspondence to J. M. Rodríguez.

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Communicated by Hossein Hajiabolhassan.

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Carballosa, W., de la Cruz, A. & Rodríguez, J.M. Gromov Hyperbolicity in the Cartesian Sum of Graphs. Bull. Iran. Math. Soc. 44, 837–856 (2018). https://doi.org/10.1007/s41980-018-0055-4

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Keywords

  • Cartesian sum of graphs
  • Geodesics
  • Gromov hyperbolicity
  • Complement of graphs

Mathematics Subject Classification

  • Primary 05C75
  • Secondary 05C12
  • 05A20