Vertex-Transitivity

Question

I was wondering if someone could help me with the following: I could see in several places (without proof) that the graph formed by the two middle layers of any boolean lattice (power set lattice) of odd rank is vertex-transitive (i.e. for any v and w there is a graph authomorphism which sends v to w)! Could anyone please tell me why this is true?

Answer 1

Here's an outline of a proof. Let $n=2k+1$; you will have two sets of vertices, $A$ and $B$, where the vertices are labelled according to $k$-subsets and $(k+1)$-subsets of $[n]$, respectively.

Given two vertices $v,w$, we must find an automorphism sending $v$ to $w$. If $v,w \in A$ or $v,w \in B$, then we choose the automorphism that permutes elements of $[n]$ as necessary. If $v \in A$ and $w \in B$, and $v$ and $w$ correspond to complementary subsets of $[n]$, then the automorphism is the complement operation on $[n]$.

Now combine the above to get the general case for $v \in A$ and $w \in B$.