Proof of the linkage conjecture for highly connected tournaments

TL;DR

This paper disproves a conjecture that highly connected tournaments with certain out-degree conditions are k-linked, by providing counterexamples, and establishes optimal conditions for semicomplete digraphs to be k-linked.

Contribution

It constructs counterexamples to the linkage conjecture for tournaments and proves an optimal linkage condition for semicomplete digraphs.

Findings

Counterexamples with minimum out-degree at least (k^2+11k)/26 for k≥42.

Proves that (2k+1)-connected semicomplete digraphs with out-degree ≥7k^2+36k are k-linked.

Refines previous results by establishing optimal out-degree conditions for k-linkage.

Abstract

A digraph $D$ is $k$-linked if for every $2k$ distinct vertices $ x_1,\ldots , x_k, y_1, \ldots , y_k$ in $D$, there exist $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ starts at $x_i$ and ends at $y_i$ for each $i\in [k]$. In 2021, Gir\~{a}o, Popielarz, and Snyder [Combinatorica 41 (2021) 815--837] conjectured that there exists a constant $C >0$ such that every $(2k+1)$-connected tournament with minimum out-degree at least $Ck$ is $k$-linked. In this paper, we disprove this conjecture by constructing a family of counterexamples with minimum out-degree at least $\frac{k^2+11k}{26}$ (for $k\geq 42$). Further, we prove that every $(2k+1)$-connected semicomplete digraph $D$ with minimum out-degree at least $ 7k^2 + 36k$ is $k$-linked. This result is optimal in terms of both connectivity and minimum out-degree (up to a multiplicative factor), which refines and generalizes the earlier result of Gir\~{a}o, Popielarz, and Snyder.