  Let $k$ be a fixed integer. We determine the complexity of finding a
$p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such
that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq
i\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show
that this problem is polynomial-time solvable when $p\geq 2k$ and ${\cal
NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question
posed in \cite{bangTCS636}. We also determine, for all fixed non-negative
integers $k_1,k_2,p$, the complexity of deciding whether a given digraph of
maximum out-degree $p$ has a $2$-partition $(V_1,V_2)$ such that the digraph
induced by $V_i$ has maximum out-degree at most $k_i$ for $i\in [2]$. It
follows from this characterization that the problem of deciding whether a
digraph has a 2-partition $(V_1,V_2)$ such that each vertex $v\in V_i$ has at
least as many neighbours in the set $V_{3-i}$ as in $V_i$, for $i=1,2$ is
${\cal NP}$-complete. This solves a problem from \cite{kreutzerEJC24} on
majority colourings.
