Consider a permutohedron $P_n$ (this is a polytope which is a convex hull of $n!$ points, which are obtained from $(1,2,...,n)$ by all possible permutations of coordinates). I have to prove the following: 2 vertices are connected with an edge iff their coordinates differ by transposition of 2 numbers which differ by 1.
I have managed to prove only one side (if they differ in such way then they are connected with an edge). Please help.