Hamiltonian path analysis

Question

I was told that to figure out if a graph had a Hamiltonian path it would not be calculated in polynomial time. Let's assume that we CAN solve it in Polynomial time, how can I prove this? is it impossible since it is NP hard problem?

Answer 1

The Hamiltonian Path Problem is NP-complete.
Nobody has proven that it is impossible to solve it in polynomial time, but it is highly unlikely that such a solution exists. If you do happen to find one, this implies that you have proven P=NP, and the Clay Math Institute will literally award you a million dollars for it :)