Infinite vertex-transitive, edge-transitive non-$1$-transitive graphs
Carsten Thomassen, Mark E. Watkins · Proceedings of the American Mathematical Society · 1989
We show that every vertex-transitive, edge-transitive graph of odd valence and subexponential growth is $1$-transitive, thus extending to infinite graphs a theorem of W. T. Tutte for finite graphs. We describe a number of counterexamples in the case of exponential growth.