  Let $G$ be a connected graph with maximum degree $\Delta \geq 3$ distinct
from $K_{\Delta + 1}$. Generalizing Brooks' Theorem, Borodin, Kostochka and
Toft proved that if $p_1, \dots, p_s$ are non-negative integers such that $p_1
+ \dots + p_s \geq \Delta - s$, then $G$ admits a vertex partition into parts
$A_1, \dots, A_s$ such that, for $1 \leq i \leq s$, $G[A_i]$ is
$p_i$-degenerate. Here we show that such a partition can be performed in linear
time. This generalizes previous results that treated subcases of a conjecture
of Abu-Khzam, Feghali and Heggernes~\cite{abu2020partitioning}, which our
result settles in full.
