On the 2-Linkage Problem for Split Digraphs

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.

Figures (8)

• Figure 1: Red diamonds represent vertices in $V_1$, blue vertices represent vertices in $V_2$. Let $P_i = s_1 \to x_i \to y_i \to z_i \to t_1$ and $Q_i = s_2 \to z_i \to y_{i+1} \to x_{i+2} \to t_2$, where subscripts in $Q_i$ are taken modulo $3$. Combined with $d_{D-\{s_2,t_2\}}^+(s_1)=3$ and $d_{D-\{s_1,t_1\}}^+(s_2)=3$, this yields $\kappa_{D\setminus\{s_2,t_2\}}(s_1,t_1)=3$ and $\kappa_{D\setminus\{s_1,t_1\}}(s_2,t_2)=3$.
• Figure 2: Example with $i_0 = 1$ and $j_0 = 3$. Red solid stars represent vertices in $V_1$, blue diamonds represent vertices in $V_2$, and black vertices are not fixed to be in either $V_1$ or $V_2$. The paths $P_1,P_2,P_3$ are directed from top to bottom. This coloring rule is used for all following figures (without further mentioning).
• Figure 3: Example with $i_0=1$ and $j_0=3$. (a) illustrates the case when $Q_1[a,t_2]$ and $P_3$ are disjoint; (b) illustrates the case when $Q_3[s_2,c]$ and $P_1$ are disjoint.
• Figure 4: Example with $i_0=1$, $j_0=3$.
• Figure 5: Example with $i_0 = 1$ and $j_0 = 3$. Red solid stars represent vertices in $V_1$, blue diamonds represent vertices in $V_2$, and black vertices are not fixed to be in either $V_1$ or $V_2$. The paths $P_1,P_2,P_3$ are directed from top to bottom. This coloring rule is used for all following figures (without further mentioning).

Theorems & Definitions (39)

• Conjecture 1.1: thomassen19802-linked
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Conjecture 1.2
• Theorem 2.1