  We generalise the standard constructions of a Cayley graph in terms of a
group presentation by allowing some vertices to obey different relators than
others. The resulting notion of presentation allows us to represent every
vertex transitive graph. As an intermediate step, we prove that every countably
infinite, connected, vertex transitive graph has a perfect matching.
Incidentally, we construct an example of a 2-ended cubic vertex transitive
graph which is not a Cayley graph, answering a question of Watkins from 1990.
