Suppose that Γ is a countably infinite group with fixed finite, symmetric generating set S of cardinality d which acts freely on the standard Borel space X. We consider the graph G on X in which two distinct points are adjacent iff an element of S sends one to the other. Improving the d+1 bound following from Kechris-Solecki-Todorcevic (1999), we show that when Γ has one end there is a Borel coloring of G using d colors (analogous to the Brooks bound for finite graphs). Time permitting, we also discuss applications to "random d-colorings" of Γ, that is, the construction of probability measures on the (compact) space of d-colorings of Γ invariant under natural automorphisms. This is joint work with Alexander Kechris.