  We demonstrate clear weak anti-localization (WAL) effect arising from induced
Rashba spin-orbit coupling (SOC) in WS$_2$-covered single-layer and bilayer
graphene devices. Contrary to the uncovered region of a shared single-layer
graphene flake, WAL in WS$_2$-covered graphene occurs over a wide range of
carrier densities on both electron and hole sides. At high carrier densities,
we estimate the Rashba SOC relaxation rate to be $\sim 0.2 \rm{ps}^{-1}$ and
show that it can be tuned by transverse electric fields. In addition to the
Rashba SOC, we also predict the existence of a `valley-Zeeman' SOC from
first-principles calculations. The interplay between these two SOC's can open a
non-topological but interesting gap in graphene; in particular, zigzag
boundaries host four sub-gap edge states protected by time-reversal and
crystalline symmetries. The graphene/WS$_2$ system provides a possible platform
for these novel edge states.
