Partitioning into degenerate graphs in linear time

Let G be a connected graph with maximum degree Δ≥3 distinct from KΔ+1. Generalizing Brooks’ Theorem, Borodin and independently Bollobás Manvel, proved that if p1,…,ps are non-negative integers such p1+⋯+ps≥Δ−s, then admits vertex partition into parts A1,…,As that, for 1≤i≤s, G[Ai] is pi-degenerate. Here we show can performed in time O(n+m). This generalizes previous results treated subcases of conjecture Abu-Khzam et al. (2020) which our result settles full.