  In this paper we study the uniqueness of graphical mean curvature flow. We
consider as initial conditions graphs of locally Lipschitz functions and prove
that in the one dimensional case solutions are unique without any further
assumptions. This result is then generalized for rotationally symmetric
solutions. In the general $n$- dimensional case, we prove uniqueness under
additional conditions: we require a { \em uniform lower bound } on the second
fundamental form and the height function of the initial condition. The latter
result extends to initial conditions that are proper graphs over subdomains of
$\mathbb{R}^n$.
