  We conduct experimental studies on the electrical transport properties of
monolayer graphene directly covered by a few layers of $\rm CrI_3$. We do not
observe the expected magnetic exchange coupling in the graphene but instead
discover proximity effects featuring gate and magnetic field tunability. The
tunability of gate voltage is manifested in the alignment of the lowest
conduction band of $\rm CrI_3$ and the Fermi level of graphene, which can be
controlled by the gate voltage. The coexistence of the normal and atypical
quantum Hall effects in our device also corresponds to gate-control modulation
doping. The lowest conduction band depends on the magnetic states of the $\rm
CrI_3$ and can be altered by the magnetic field, which corresponds to the
resistance loops during back-and-forth sweeps of the magnetic field. Our
results serve as a reference for exploiting the magnetic proximity effects in
graphene.
