On cubic vertex-transitive graphs of given girth

A set of vertices of a graph is distinguishing if the only automorphism that preserves it is the identity. The minimal size of such sets, if they exist, is the distinguishing cost. The distinguishing costs of vertex transitive cubic graphs are well known if they are 1-arc-transitive, or if they have two edge orbits and either have girth 3 or vertex-stabilizers of order 1 or 2. There are many results about vertex-transitive cubic graphs of girth 4 with two edge orbits, but for larger girth almost nothing is known about %the existence or the distinguishing costs of such graphs. We prove that cubic vertex-transitive graphs of girth 5 with two edge orbits have distinguishing cost 2, and prove the non-existence of infinite 3-arc-transitive cubic graphs of girth 6.