Partitioning into degenerate graphs in linear time

Abstract

Let G be a connected graph with maximum degree ∆ ≥ 3 distinct from K∆+1. Generalizing Brooks' Theorem, Borodin, Kostochka and Toft proved that if p1, . . . , ps are non-negative integers such that p1 +• • • +ps ≥ ∆−s, then G admits a vertex partition into parts A1, . . . , As such that, for 1 ≤ i ≤ s, G[Ai] is pi-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 [2], which our result settles in full.