  Enhancing the spin-orbit interaction in graphene, via proximity effects with
topological insulators, could create a novel 2D system that combines nontrivial
spin textures with high electron mobility. In order to engineer practical
spintronics applications with such graphene/topological insulator (Gr/TI)
heterostructures, an understanding of the hybrid spin-dependent properties is
essential. {However to date, despite the large number of experimental studies
on Gr/TI heterostructures reporting a great variety of remarkable (spin)
transport phenomena, little is known about the true nature of the spin texture
of the interface states as well as their role on the measured properties. Here
we use {\it ab initio} simulations and tight-binding models to determine the
precise spin texture of electronic states in graphene interfaced with a
Bi$_2$Se$_3$ topological insulator. Our calculations predict the emergence of a
giant spin lifetime anisotropy in the graphene layer, which should be a
measurable hallmark of spin transport in Gr/TI heterostructures, and suggest
novel types of spin devices
