Citation:
Carballosa, W., de la Cruz, A., Martínez-Pérez, A., Rodríguez, J.M. (2018). Hyperbolicity of Direct Products of Graphs. Symmetry, 10 (7), 279.
Sponsor:
This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain.
Project:
Gobierno de España. MTM2012-30719 Gobierno de España. MTM2013-46374-P Gobierno de España. MTM2016-78227-C2-1-P Gobierno de España. MTM2015-69323-REDT
Keywords:
Direct product of graphs
,
Geodesics
,
Gromov hyperbolicity
,
Bipartite graphs
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 hyperbolicitIt 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 G(1) x G(2) 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).[+][-]