  If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it 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 $\delta$-\emph{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\ge 0: \, X \,
\text{ is $\delta$-hyperbolic}\,\}.$ Some previous works characterize the
hyperbolic product graphs (for the Cartesian, strong, join, corona and
lexicographic products) in terms of properties of the factor graphs. However,
the problem with the direct product is more complicated. In this paper, we
prove that if the direct product $G_1\times G_2$ is hyperbolic, then one factor
is hyperbolic and the other one is bounded. Also, we prove that this necessary
condition is, in fact, a characterization in many cases. In other cases, we
find characterizations which are not so simple. Furthermore, we obtain formulae
or good bounds for the hyperbolicity constant of the direct product of some
important graphs.
