On Complete Constant Mean Curvature Vertical Multigraphs in E(κ, τ)

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-2 X R) is either a vertical cylinder or a vertical graph (in both cases, it must be 4H(2) + kappa <= 0).