On the $k$-linkage problem for generalizations of semicomplete digraphs

TL;DR

The paper addresses the $k$-linkage problem in a broad digraph class that includes semicomplete and quasi-transitive digraphs, adapting Thomassen’s linkage framework beyond tournaments. It introduces a robust linking strategy built on nearly in-dominating sets and anchoring lemmas, and proves that every $3k$-strong digraph in the class with $\delta^+(D)\geq 23k$ is $k$-linked, with extensions to semicomplete compositions and to $l$-quasi-transitive digraphs via a function $f(k,l)$. The main technical advance strengthens properties of in-degree-maximal vertices to yield abundant short, independent $(v,u)$-paths, enabling Menger-type constructions that guarantee disjoint $x_i\to y_i$ paths. These results broaden the scope of Thomassen-type linkage results and provide a cohesive framework for handling linkage in semicomplete compositions and related digraph classes, with implications for understanding connectivity versus linkage in non-tournament settings.

Abstract

A directed graph (digraph) $ D $ is $ k $-linked if $ |D| \geq 2k $, and for any $ 2k $ distinct vertices $ x_1, \ldots, x_k, y_1, \ldots, y_k $ of $ D $, there exist vertex-disjoint paths $ P_1, \ldots, P_k $ such that $ P_i $ is a path from $ x_i $ to $ y_i $ for each $ i \in [k] $. In 1980, Thomassen conjectured that there exists a function $ f(k) $ such that every $ f(k) $-strong digraph is $ k $-linked. He later disproved this conjecture by showing that $ f(2) $ does not exist for general digraphs and proved that the function $f(k)$ exists for the class of tournaments. In this paper we consider a large class $\mathcal{ D} $ of digraphs which includes all semicomplete digraphs (digraphs with no pair of non-adjacent vertices) and all quasi-transitive digraphs (a digraph $D$ is quasi-transitive if for any three vertices $x, y, z$ of $D$, whenever $xy$ and $yz$ are arcs, then $x$ and $z$ are adjacent). We prove that every $ 3k $-strong digraph $D\in \mathcal{D}$ with minimum out-degree at least $ 23k $ is $ k $-linked. A digraph $D$ is $l$-quasi-transitive if whenever there is a path of length $l$ between vertices $u$ and $v$ in $D$ the vertices $u$ and $v$ are adjacent. Hence 2-quasi-transitive digraphs are exactly the quasi-transitive digraphs. We prove that there is a function $f(k,l)$ so that every $f(k,l)$-strong $l$-quasi-transitive digraph is $k$-linked. The main new tool in our proofs significantly strengthens an important property of vertices with maximum in-degree in a tournament. While Landau in 1953 already proved that such a vertex $v$ is reachable by all other vertices by paths of length at most 2, we show that, in fact, the structure of these paths is much richer. In general there are many such paths for almost all out-neighbours of $v$ and this property is crucial in our proofs.

Figures (2)

• Figure 1: The basic structure of the linking paths for $k=4$.
• Figure 2: The construction of $D'$: The dashed lines indicate that these arcs are absent in $D$. We add arcs in $D'$ in the direction indicated by the arrows. Specifically, an arc $uv$ is added to $D'_0$ if and only if there exist ${f(k,l)}$ independent $(u,v)$-paths in $D_0$.

Theorems & Definitions (27)

• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.1
• Theorem 1.2
• Corollary 1.1
• Theorem 1.3
• Corollary 1.2
• Theorem 1.4
• Theorem 2.1
• Definition 2.1