Automorphism groups of vertex transitive graphs Does there exist a finite nonoriented graph whose automorphism group is transitive but not generously transitive (that is, it is not true that each pair $(x,y)$ of vertices can be interchanged by some automorphism)?
I know, for example, that Cayley graphs are always generously vertex-transitive.

To provide a more explicit example: Take two vertices of distance 2 on the truncated tetrahedron. Observe that there is a unique path between these vertices, and that the edges in this path are distinct (only one of them is part of a triangle, for instance). So we can define a direction on this path, and thus can't hope to flip it in an automorphism.
