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Symmetry 2018, 10(7), 279; https://doi.org/10.3390/sym10070279

Hyperbolicity of Direct Products of Graphs

1
Department of Mathematics and Statistics, Florida International University, 11200 SW 8th Street, Miami, FL 33199, USA
2
Department of Mathematics, Miami Dade College, 300 NE Second Ave. Miami, FL 33132, USA
3
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, 28911 Madrid, Spain
4
Facultad CC. Sociales de Talavera, Universidad de Castilla La Mancha, Avda. Real Fábrica de Seda, s.n. Talavera de la Reina, 45600 Toledo, Spain
*
Author to whom correspondence should be addressed.
Received: 11 June 2018 / Revised: 3 July 2018 / Accepted: 9 July 2018 / Published: 12 July 2018
(This article belongs to the Special Issue Symmetry in Graph Theory)
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Abstract

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs). View Full-Text
Keywords: direct product of graphs; geodesics; Gromov hyperbolicity; bipartite graphs direct product of graphs; geodesics; Gromov hyperbolicity; bipartite graphs
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Carballosa, W.; de la Cruz, A.; Martínez-Pérez, A.; Rodríguez, J.M. Hyperbolicity of Direct Products of Graphs. Symmetry 2018, 10, 279.

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