Partitioning into degenerate graphs in linear time

Abstract

Let $G$ be a connected graph with maximum degree $\Delta \ge 3$ 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 $p_1+\cdots +p_s\ge \Delta -s$, then $G$ admits a vertex partition into parts $A_1,\dots ,A_s$ such that, for $1\le i\le s$, $G[A_i]$ 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.

12 pages, 4 figures, 5 algorithms