Gromov hyperbolicity in strong product graphs

If X is a geodesic metrics pace and x(1), x(2), x(3) is an element of 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 delta-hyperbolic (in the Gromov sense) if any side of T is contained in a delta-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by delta(X) the sharp hyperbolicity constant of X, i.e. delta(X) = inf{delta >= 0 : X is delta-hyperbolic}. In this paper we characterize the strong product of two graphs G(1) boxed times G(2) which are hyperbolic, in terms of G(1) and G(2): the strong product graph G(1) boxed times G(2) is hyperbolic if and only if one of the factors is hyperbolic and the other one is bounded. We also prove some sharp relations between delta(G(1) boxed times G(2)), delta(G(1)), delta(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 exact values of the hyperbolicity constant for many strong product graphs.

Gromov hyperbolicity in strong product graphs Articles