Gromov hyperbolicity in Cartesian product graphs (original) (raw)
2010, Proceedings - Mathematical Sciences
If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) = inf{δ ≥ 0: X is δ-hyperbolic}. In this paper we characterize the product graphs G 1 × G 2 which are hyperbolic, in terms of G 1 and G 2 : the product graph G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic and G 2 is bounded or G 2 is hyperbolic and G 1 is bounded. We also prove some sharp relations between the hyperbolicity constant of G 1 × G 2 , δ(G 1 ), δ(G 2 ) and the diameters of G 1 and G 2 (and we find families of graphs for which the inequalities are attained). Furthermore, we obtain the precise value of the hyperbolicity constant for many product graphs.
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