  We investigate the electronic band structure of graphene on a series of
two-dimensional magnetic transition-metal phosphorus trichalcogenide
monolayers, MPX$_3$ with M={Mn,Fe,Ni,Co} and X={S,Se}, with first-principles
calculations. A symmetry-based model Hamiltonian is employed to extract orbital
parameters and sublattice resolved proximity-induced exchange couplings
($\lambda_{\textrm{ex}}^\textrm{A}$ and $\lambda_{\textrm{ex}}^\textrm{B}$)
from the low-energy Dirac bands of the proximitized graphene. Depending on the
magnetic phase of the MPX$_3$ layer (ferromagnetic and three antiferromagnetic
ones), completely different Dirac dispersions can be realized with exchange
splittings ranging from 0 to 10~meV. Surprisingly, not only the magnitude of
the exchange couplings depends on the magnetic phase, but also the global sign
and the type. Important, one can realize uniform
($\lambda_{\textrm{ex}}^\textrm{A} \approx \lambda_{\textrm{ex}}^\textrm{B}$)
and staggered ($\lambda_{\textrm{ex}}^\textrm{A} \approx
-\lambda_{\textrm{ex}}^\textrm{B}$) exchange couplings in graphene. From
selected cases, we find that the interlayer distance, as well as a transverse
electric field are efficient tuning knobs for the exchange splittings of the
Dirac bands. More specifically, decreasing the interlayer distance by only
about 10\%, a giant 5-fold enhancement of proximity exchange is found, while
applying few V/nm of electric field, provides tunability of proximity exchange
by tens of percent. We have also studied the dependence on the Hubbard $U$
parameter and find it to be weak. Moreover, we find that the effect of SOC on
the proximitized Dirac dispersion is negligible compared to the exchange
coupling.
