Partitioning into degenerate graphs in linear time

TL;DR

This paper proves that a specific vertex partitioning of graphs into degenerate subgraphs, generalizing Brooks' Theorem, can be achieved in linear time, settling a longstanding conjecture.

Contribution

It introduces a linear-time algorithm for partitioning graphs into degenerate parts, extending previous partial results and fully resolving a conjecture.

Findings

Partitioning into degenerate graphs can be done in linear time.

Generalizes Brooks' Theorem to broader graph classes.

Settles a conjecture by Abu-Khzam, Feghali, and Heggernes.

Abstract

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.