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Characterization of Gromov hyperbolic short graphs

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Abstract

To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S 9r (G) is finite, where S R (G):= sup{L(C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d C (x, y) ≤ d G (x, y) + R for every x, y ∈ C.

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  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, Madrid, 28911, Spain

    José Manuel Rodríguez

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  1. José Manuel Rodríguez
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Correspondence to José Manuel Rodríguez.

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Supported by Ministerio de Ciencia e Innovación of Spain (Grant No. MTM 2009-07800), and a grant from Consejo Nacional De Ciencia Y Tecnologia of México (Grant No. CONACYT-UAG I0110/62/10)

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Rodríguez, J.M. Characterization of Gromov hyperbolic short graphs. Acta. Math. Sin.-English Ser. 30, 197–212 (2014). https://doi.org/10.1007/s10114-013-2467-7

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  • DOI: https://doi.org/10.1007/s10114-013-2467-7

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