On the 2-Linkage Problem for Split Digraphs

Xiaoying Chen; Jørgen Bang-Jensen; Jin Yan; Jia Zhou

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arXiv · 2603.07603

On the 2-Linkage Problem for Split Digraphs

Xiaoying Chen·Jørgen Bang-Jensen·Jin Yan·Jia Zhou

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Abstract

A digraph is {\bf $ k $-linked} if for arbitary two disjoint vertex sets $\{s_1, \ldots, s_k\}$ and $\{t_1, \ldots, t_k\}$, there exist vertex-disjoint directed paths $P_1, \ldots, P_k$ {such that $P_i$ is a directed path from $s_i$ to $t_i$ for each $i\in [k]$}. A {\bf split digraph} is a digraph $ D = (V_1, V_2; A) $ whose vertex set is a disjoint union of two nonempty sets $ V_1 $ and $ V_2 $ such that $ V_1 $ is an independent set and the subdigraph induced by $ V_2 $ is semicomplete (no pair of non-adjacent vertices). A {\bf semicomplete split digraph} is a split digraph $ D = (V_1, V_2; A) $ in which every vertex in the independent set $ V_1 $ is adjacent to every vertex in $ V_2 $. {Semicomplete split digraphs form an important subclass of the class of semicomplete multipartite digraphs.} In this paper, we prove that every 6-strong split digraph is 2-linked. This solves a problem posed by Bang-Jensen and Wang [J. Graph Theory, 2025]. We also show that every 5-strong semicomplete split digraph is 2-linked. This bound is tight already for semicomplete digraphs.

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BibTeX RIS

Xiaoying Chen, Jørgen Bang-Jensen, Jin Yan, Jia Zhou. 2026-03-08. On the 2-Linkage Problem for Split Digraphs. https://arxiv.org/abs/2603.07603

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