On the 2-Linkage Problem for Split Digraphs

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.