On the 2-Linkage Problem for Split Digraphs

TL;DR

This paper proves that 6-strong split digraphs are 2-linked, solving a longstanding problem, and shows that 5-strong semicomplete split digraphs are also 2-linked, with tight bounds established.

Contribution

It establishes new linkage properties for split digraphs, specifically proving 6-strong split digraphs are 2-linked and 5-strong semicomplete split digraphs are 2-linked.

Findings

Every 6-strong split digraph is 2-linked.

Every 5-strong semicomplete split digraph is 2-linked.

The bound for semicomplete digraphs is tight.

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.