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.

翻译：令 $G$ 为最大度 $\Delta \geq 3$ 且不同于 $K_{\Delta + 1}$ 的连通图。推广Brooks定理，Borodin、Kostochka与Toft证明：若 $p_1, \dots, p_s$ 为非负整数且满足 $p_1 + \dots + p_s \geq \Delta - s$，则 $G$ 存在顶点划分 $A_1, \dots, A_s$，使得对任意 $1 \leq i \leq s$，子图 $G[A_i]$ 为 $p_i$-退化。本文证明此类划分可在线性时间内完成。该结果推广了针对Abu-Khzam、Feghali与Heggerne猜想（文献~\cite{abu2020partitioning}）子情形的既有结论，并完全解决了该猜想。