Arc-disjoint in- and out-branchings in semicomplete split digraphs

Abstract

An out-tree (in-tree) is an oriented tree where every vertex except one, called the root, has in-degree (out-degree) one. An out-branching $B_u^{+}$ (in-branching $B_u^{-}$) of a digraph $D$ is a spanning out-tree (in-tree) rooted at $u$. A good $(u,v)$-pair in $D$ is a pair of branchings $B_u^{+},B_v^{-}$ which are arc-disjoint. Thomassen proved that deciding whether a digraph has any good pair is NP-complete. A semicomplete split digraph is a digraph where the vertex set is the disjoint union of two non-empty sets, $V_1$ and $V_2$, such that $V_1$ is an independent set, the subdigraph induced by $V_2$ is semicomplete, and every vertex in $V_1$ is adjacent to every vertex in $V_2$. In this paper, we prove that every 2-arc-strong semicomplete split digraph $D$ contains a good $(u,v)$-pair for any choice of vertices $u,v$ of $D$, thereby confirming a conjecture by Bang-Jensen and Wang [Bang-Jensen and Wang, J. Graph Theory, 2024].