A tournament is defined as an orientation of a complete graph. Prove or disprove the following statement:
In a tournament, there are exactly an odd number of Hamiltonian paths.
In all cases I’ve tried, the statement seems true, so I guess it’s always true. But to prove it, I really got confused. Two ideas have come up but get nowhere. Please help.
- Induction on number of vertices;
- Show that changing the direction of one edge won’t change the parity.