Oriented Hamiltonian Paths in Tournaments: Stability under Arc Deletion

Havet and Thomassé 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.

💡 Research Summary

This paper investigates the robustness of a fundamental property in tournament graph theory: the containment of all oriented Hamiltonian paths. The foundational result, established by Havet and Thomassé, confirms Rosenfeld’s conjecture that every tournament of order n ≥ 8 contains every possible oriented Hamiltonian path. The present work extends this by asking: if we delete a single arbitrary arc from the tournament, does this property still hold?

The authors prove a strong stability result. For any tournament T of order n ≥ 8 and for any arc e = (x, y), the digraph D = T - e obtained by deleting e still contains every oriented Hamiltonian path, with only a few explicitly characterized exceptions. These exceptional cases, termed “special exceptions,” are precisely defined. There are two types: (1) when the path P is directed, the vertices x and y have identical in-neighborhoods and identical out-neighborhoods in D, and the in-neighborhood dominates the out-neighborhood; (2) when P has exactly two blocks, and both x and y become sinks (vertices with out-degree zero) in D.

The proof is a sophisticated case analysis that leverages the powerful theorem of Havet and Thomassé (Theorem 1) as its primary engine. This theorem provides a condition based on the “out-section” of two vertices to guarantee that one of them is an origin of a given oriented path. The authors skillfully apply this tool to the modified graph D. A key intermediate step (Lemma 1) handles situations where the subtournament T_x = T - x falls into one of the known exception categories from prior work, showing that in most such subcases, D still contains the desired path P.

The main proof proceeds by assuming, without loss of generality, that P is an out-path. It first disposes of trivial cases (e.g., when x has no out-neighbors in D). The core of the argument focuses on the case where the first block of P has length at least 2. Here, the proof branches based on whether P is directed or not, and further subdivides based on the out-degree of x in D. The authors meticulously demonstrate that unless the configuration matches one of the two defined special exceptions, a Hamiltonian path isomorphic to P can always be constructed in D. This involves considering whether certain subgraphs are exceptions, using induction-like arguments on path blocks, and constructing the path by cleverly inserting the vertices x and y into a path found in a smaller subgraph.

In conclusion, this paper provides a complete characterization of when the deletion of a single arc preserves the universal existence of oriented Hamiltonian paths in sufficiently large tournaments. It confirms that this property is highly robust, failing only in very specific, well-understood structural configurations. This work deepens the understanding of Hamiltonian properties in tournaments and contributes to the broader study of fault tolerance in network structures.