  Graphene-based heterostructures are a promising material system for designing
the topologically nontrivial Chern insulating devices. Recently, a
two-dimensional (2D) monolayer ferromagnetic insulator CrI$_{3}$ was
successfully synthesized in experiments [Huang et al., Nature 546, 270 (2017)].
Here, these two interesting materials are proposed to build a heterostructure
(Gr/CrI$_{3}$). Our first-principles calculations show that the system forms a
van der Waals (vdW) heterostructure, relatively facilely fabricated in
experiments. A Chern insulating state is acquired in the Gr/CrI$_{3}$
heterostructure if the vdW gap is compressed to certain extents by applying an
external pressure. Amazingly, very strong magnetization (about 150 meV) is
found in graphene, induced by the substrate CrI$_{3}$, despite the vdW
interactions between them. A low-energy effective model is employed to
understand the mechanism. The work functions, contact types, and band
alignments of the Gr/CrI$_{3}$ heterostructure system are also studied. Our
work demonstrates that the Gr/CrI$_{3}$ heterostructure is a promising system
to observe the quantum anomalous Hall effect at high temperatures (up to 45 K)
in experiments.
