Oriented Hamiltonian Paths in Tournaments: Stability under Arc Deletion

TL;DR

This paper investigates the robustness of the property that tournaments contain all oriented Hamiltonian paths, showing that this property generally persists after deleting any arc, with some explicitly characterized exceptions.

Contribution

It extends previous results by proving that in large tournaments, the property of containing all oriented Hamiltonian paths remains stable under any arc deletion, except for specific known cases.

Findings

Most arcs in large tournaments preserve the Hamiltonian path property after deletion.

Explicit exceptions where the property fails are fully characterized.

The stability result applies to tournaments of order at least 8.

Abstract

Havet and Thomass\'{e} proved that every tournament of order $n\geq 8$ contains every oriented Hamiltonian path, which was conjectured by Rosenfeld. Recently, it was shown that in any tournament $T$ of order $n\geq 8$, 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 $n\geq 8$ satisfies that $T-e$ contains every oriented Hamiltonian path, except for some explicitly described exceptions.