  The low temperature phase diagram of $^4$He adsorbed on a single graphene
sheet is studied by computer simulation of a system comprising nearly thousand
helium atoms. In the first layer, two commensurate solid phases are observed,
with fillings 1/3 and 7/16 respectively, separated by a domain wall phase, as
well as an incommensurate crystal at higher coverage. No evidence of a
thermodynamically stable superfliuid phase is found for the first adlayer.
Second layer promotion occurs at a coverage of 0.111(4) $\AA^{-2}$. In the
second layer two phases are observed, namely a superfluid and an incommensurate
solid, with no commensurate solid intervening between these two phases. The
computed phase diagram closely resembles that predicted for helium on graphite.
