  Transition metal thriclorides, with $4d$ or $5d$ electrons, are materials at
the forefront of recent studies about the interplay of spin-orbit coupling and
strong Coulomb interactions. Within our first-principles calculations
(DFT+$U$+SOC) we study the effects of graphene on the electronic and magnetic
properties of the monolayers of $\alpha$-RuCl$_3$ and OsCl$_3$. Despite the
spatially inhomogeneous $n$-type doping induced by graphene, we show that the
occupancy of the upper Hubbard bands of MLs of \rucl and OsCl$_3$ can be tuned
through external electric fields, and allows the control of (i) metal-insulator
transitions, and (ii) the magnetic easy-axis and anisotropy energies. Our
findings point towards the tunning of electronic and magnetic properties of
transition metal thriclorides monolayers by using graphene and external
electronic fields.
