  The phase diagram of the first layer of $^4$He adsorbed on a single graphene
sheet has been calculated by a series of diffusion Monte Carlo calculations
including corrugation effects. As the number of C-He interactions is reduced
with respect to graphite the binding energy of $^4$He atoms to graphene is
approximately 13.4 K/He atom smaller. Our results indicate that the phase
diagram is qualitatively similar to that of helium on top of graphite. A
two-dimensional liquid film on graphene is predicted to be metastable with
respect to the commensurate solid but the difference in energy between both
phases is very small, opening the possibility of such a liquid film to be
experimentally observed.
