Let X be a graph and let G be a subgroup of the automorphism group of X. We investigate the question of when the full 2-shift process on the vertices of X has a G-factor process on the same graph consisting of random variables which are independent and identically distributed (i.i.d.) uniform in [0, 1]. We show that such a factor always exists when X is a tree with bounded degrees, no leaves and at least three topological ends, and give a sufficient condition for when such a factor exists if X is a connected graph with infinitely many ends and G acts transitively on the vertices of X. We also show that the full 2m-shift on any finitely-generated non-abelian group G has a G-factor which is i.i.d. uniform on [0, 1] when m is sufficiently large.