  We prove that any complete surface with constant mean curvature in a
homogeneous space E(\kappa,\tau) which is transversal to the vertical Killing
vector field is, in fact, a vertical graph. As a consequence we get that any
orientable, parabolic, complete, immersed surface with constant mean curvature
H in E(\kappa,\tau) (different from a horizontal slice in S^2xR) is either a
vertical cylinder or a vertical graph (in both cases, it must be
4H^2+\kappa\leq0).
