Summary: If \(X\) is a geodesic metric space and \(x_1,x_2,x_3 \in X\), a geodesic triangle \(T = \{x_1,x_2,x_3\}\) is the union of the three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) 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., \(\delta(X) = \inf\{\delta \geq 0 : X\text{ is }\delta\text{-hyperbolic}\}\).
In this paper we characterize the product graphs \(G_1 \times G_2\) which are hyperbolic, in terms of \(G_1\) and \(G_2\): the product graph \(G_1\times G_2\) 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 \(G_1 \times G_2\), \(\delta(G_1)\), \(\delta(G_2)\) 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 precise value of the hyperbolicity constant for many product graphs.