  Schreier graphs, which possess both a graph structure and a Schreier
structure (an edge-labeling by the generators of a group), are objects of
fundamental importance in group theory and geometry. We study the Schreier
structures with which unlabeled graphs may be endowed, with emphasis on
structures which are invariant in some sense (e.g. conjugation-invariant, or
sofic). We give proofs of a number of "folklore" results, such as that every
regular graph of even degree admits a Schreier structure, and show that, under
mild assumptions, the space of invariant Schreier structures over a given
invariant graph structure is very large, in that it contains uncountably many
ergodic measures. Our work is directly connected to the theory of invariant
random subgroups, a field which has recently attracted a great deal of
attention.
