Oriented Hamiltonian Paths in Tournaments: Stability under Arc Deletion

Havet and Thomassé proved that every tournament of order $n\ge 8$ contains every oriented Hamiltonian path, which was conjectured by Rosenfeld. Recently, it was shown that in any tournament $T$ of order $n\ge 8$, there exists an arc $e$ such that $T-e$ contains any oriented Hamiltonian path. A natural extension of this problem is to study the stability of this property under arbitrary arc deletion. In this paper, we prove that every arc $e$ in a tournament $T$ of order $n\ge 8$ satisfies that $T-e$ contains every oriented Hamiltonian path, except for some explicitly described exceptions.