[Paper Review] Highly linked tournaments

Alexey Pokrovskiy|BIROn (Birkbeck, University of London)|Jun 29, 2014

Advanced Graph Theory Research6 references4 citations

TL;DR

This paper proves that every strongly 452k-connected tournament is k-linked, confirming a conjecture by K"uhn, Lapinskas, Osthus, and Patel. It extends the linkage structure method using transitive subtournaments and Menger's Theorem to construct vertex-disjoint paths between specified vertex pairs, achieving a linear bound on connectivity for k-linked tournaments.

ABSTRACT

A (possibly directed) graph is $k$-linked if for any two disjoint sets of vertices $\{x_1, \dots, x_k\}$ and $\{y_1, \dots, y_k\}$ there are vertex disjoint paths $P_1, \dots, P_k$ such that $P_i$ goes from $x_i$ to $y_{i}$. A theorem of Bollobás and Thomason says that every $22k$-connected (undirected) graph is $k$-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás-Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments - there is a function $f(k)$ such that every strongly $f(k)$-connected tournament is $k$-linked. The bound on $f(k)$ was reduced to $O(k \log k)$ by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly $452k$-connected tournament is $k$-linked.

Motivation & Objective

• To resolve the conjecture that a linear connectivity bound suffices for tournaments to be k-linked.
• To improve upon the previous O(k log k) bound by K"uhn et al. for strong connectivity implying k-linkedness in tournaments.
• To establish a tight, explicit constant (452) for the linear connectivity threshold ensuring k-linkedness in tournaments.
• To extend the linkage structure method to achieve precise path constructions in highly connected tournaments.

Proposed method

• Constructs linkage structures using transitive subtournaments (D⁻_i, D⁺_i) of size 2 to form path components.
• Applies Menger's Theorem to find 5k vertex-disjoint paths Q_i from Y⁻ to X⁺ in T\X, where |X| ≤ 222k.
• Uses degree conditions and averaging arguments to identify additional out-neighbors and in-neighbors outside critical sets E⁻_i and E⁺_i.
• Replaces initial path endpoints with vertices outside domination sets to ensure disjointness and path continuity.
• Employs Lemma 2.1 to find disjoint paths P⁻_i and P⁺_i connecting Q⁻_i and Q⁺_i to the Q′_i paths in T⁻ and T⁺.
• Concatenates paths: x_i → x′_i → Q⁻_i → P⁻_i → Q′_i → P⁺_i → Q⁺_i → y′_i → y_i to form the required k disjoint x_i-y_i paths.

Experimental results

Research questions

• RQ1 Is there a linear function f(k) such that every strongly f(k)-connected tournament is k-linked?
• RQ2 Can the O(k log k) connectivity bound for k-linked tournaments be improved to a linear bound?
• RQ3 What is the best explicit constant C such that every strongly Ck-connected tournament is k-linked?
• RQ4 Can the linkage structure method be refined to achieve a tight linear bound in tournament k-linkedness?

Key findings

• Every strongly 452k-connected tournament is k-linked, confirming the conjecture of K"uhn, Lapinskas, Osthus, and Patel.
• The bound of 452k is explicit and constructive, improving upon the previous O(k log k) bound of 10⁴k log k.
• The proof relies on careful selection of vertices outside domination sets to ensure path disjointness and connectivity.
• The method uses transitive subtournaments and Menger’s Theorem to construct disjoint paths between specified pairs.
• The construction ensures that x′_i and y′_i can be chosen outside E⁻_i and E⁺_i with at least 2k choices, enabling disjointness.
• The final path concatenation ensures vertex-disjointness across all k paths from x_i to y_i, satisfying k-linkedness.