Groups and Graphs: Proving Transitive Action on Vertices

[Mess10]

Hi.

Need help with following problem:

Let R=(V,E) a regular graph with degree at least 1 and odd number of vertices.
Let C=Aut(R) the transitive action on the set E of R.

Prove C also transitive action on the set V of R.


Anyone got any idea/tips?

Thanks!

 

[Geomancer]

Try assuming the contrary: that the action is not transitive on V. Deduce that R is bipartite. Contradiction. Am I missing something?