  Motivated by the possible charge ordering in the recently discovered
superconductor Na_xCoO_2 yH_2O, which at filling x=1/3 would correspond to a
half-filled honeycomb lattice, we investigate the single-hole dynamics of the
t-J model on this lattice. Using exact diagonalization, series expansion and
the self-consistent Born approximation, we calculate the quasi-particle
dispersion, bandwidth and residues and compare our findings with the well
established results for the square lattice. Given the similarities between both
lattices, it is not surprising that we find qualitatively similar features for
the honeycomb lattice, namely an almost flat band along the edge of the
magnetic Brillouin zone and well defined hole pockets around the corners.
However, we report a considerable disagreement between the three methods used
concerning the bandwidth and discuss possible origins of this discrepancy.
