  The phase diagram of the first layer of H$_2$ adsorbed on top of a single
graphene sheet has been calculated by means of a series of diffusion Monte
Carlo (DMC) simulations. We have found that, as in the case of $^4$He, the
ground state of molecular hydrogen is a $\sqrt3 \times \sqrt3$ commensurate
structure, followed, upon a pressure increase, by an incommensurate triangular
solid. A striped phase of intermediate density was also considered, and found
lying on top of the equilibrium curve separating both commensurate and
incommensurate solids.
