Number of Hamilton Cycles in Complete Graph
Joining either end of that path gives us a Hamilton cycle.
Hence there are $n!$ ways of building such a Hamilton cycle.
Not all these are different, though.
On any such cycle, there are:
- $n$ different places you can start;
- $2$ different directions you can travel.
So any one of these $n!$ cycles is in a set of $2n$ cycles which all contain the same set of edges.