I'm trying to proof the following theorem:
Between every two vertices $u,v$ of a de bruijn graph $K_{2,n}$ with $n$ vertices, there is only one path of $n-1$ length.
First I wasn't sure if this theorem is true so I tried to find an example to prove it's false, but without success. I guess this theorem is correct so I tried to remove one of the vertex $v$ and play with it, but also, without any success.
How can I prove this theorem?
Ok so if you do not specify what type of path the theorem is outright false because you can loop in different places repeatedly to achieve the same number of steps in multiple ways. If you limit to only hitting each point once or each line once along the way it might be true. Removing one vertex is not a useful way to find a proof, because that limits the options and might remove a path of length n-1 between two vertices that would have existed, or it might remove both.