On mutually independent hamiltonian paths

Let P1=〈v1,v2,v3,…,vn〉P1=〈v1,v2,v3,…,vn〉 and P2=〈u1,u2,u3,…,un〉P2=〈u1,u2,u3,…,un〉 be two hamiltonian paths of GG. We say that P1P1 and P2P2 are independent   if u1=v1,un=vnu1=v1,un=vn, and ui≠viui≠vi for 1