Out-degree reducing partitions of digraphs

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 $V_i$, ($1\leq i\leq p$)isatleast$k$smallerthanthemaximumout-degreeof$D$. We show that this problem is polynomial-time solvable when $p\ge 2k$ and ${\cal NP}$-completeotherwise\mathrm{.}Theresultfor$k=1$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 $\mathcal{N}\mathcal{P}$-complete. This solves a problem from \cite{kreutzerEJC24} on majority colourings.