Is the disjoint union of 2 copies of the complete bipartite graphs vertex transitive? Is the disjoint union of $K_{n/4,n/4}$ and $K_{n/4,n/4}$ a vertex transitive graph?
I think it is true, but since I failed to come up with a proof I have some doubts about it.
Thanks
 A: It is vertex transitive, Let $A,B$ and $C,D$ be the two pairs of parts of the graph. any function that sends the vertices of one part to vertices which are all in one part and which satisfies that parts that where initially in the same pair go to parts that are still paired up is an automorphism.
Let $v$ be a vertex in part $A$ without loss of generality. We have to prove that if $u$ is another vertex then there is an automorphism that sends $v$ tu $u$.
there are four cases:
if $u$ is in $A$ then just send vertex $v$ to vertex $u$ and vertex $u$ to vertex $v$ and fix every other vertex.
If $u$ is in $B$ then send all of the vertices in $A$ to vertices in $B$ and all of the vertices in $B$ to vertices in $A$ making sure that vertex $v$ goes to vertex $u$. Don't move the vertices of $C$ and $D$.
If $u$ is in $C$ then send the vertices of $A$ to vertices in $C$, and send the vertices in $C$ to vertices in $A$, making sure $u$ goes to $v$. And then send the vertices in $B$ to vertices in $D$ and vertices in $D$ to vertices in $B$.
If $u$ is in $D$ it is analogous to when $v$ is in $C$.
