Partitioning into degenerate graphs in linear time

Abstract

Let G be a connected graph with maximum degree Δ≥3 distinct from KΔ+1. Generalizing Brooks’ Theorem, Borodin and independently Bollobás and Manvel, proved that if p1,…,ps are non-negative integers such that p1+⋯+ps≥Δ−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 time O(n+m). This generalizes previous results that treated subcases of a conjecture of Abu-Khzam et al. (2020) which our result settles in full.