This NP-complete problem requires finding a Hamiltonian circuit in a graph whose edge weights have a minimum sum. Ore's theorem gives a sufficient, but not necessary condition. Consider any circuit graph with 5 or more vertices. Let G be a graph with at least 3 vertices. There are corresponding theorems for the existence of a Hamiltonian cycle in digraphs.
Every tournament has a Hamiltonian path.
By induction on the number of vertices. If there is more than one Hamiltonian path in a tournament, the vertices do not have a unique ranking. A tournament has a unique Hamiltonian path if and only if the tournament is transitive. Let u and v be the vertices on these paths where the two paths first differ. This is a contradiction since D is a tournament.