Partitioning a graph into degenerate subgraphs

Let G = (V, E) be a connected graph with maximum degree k >= 3 distinct from Kk+1. Given integers s >= 2 and p(1), ..., p(s) >= 0, G is said to be (p(1), ..., p(s))-partitionable if there exists a partition of V into sets V-1, ..., V-s such that G[V-i] is p(i)-degenerate for i is an element of { 1, ..., s}. In this paper, we prove that we can find a (p(1), ..., p(s))-partition of G in O(vertical bar V vertical bar + vertical bar E vertical bar)-time whenever 1 >= p(1), ..., p(s) >= 0 and Sigma(s)(i=1), p(i) >= k - s. This generalizes a result of Bonamy et al. (2017) and can be viewed as an algorithmic extension of Brooks' Theorem and several results on vertex arboricity of graphs of bounded maximum degree. We also prove that deciding whether G is (p, q)-partitionable is NP-complete for every k >= 5 and pairs of non-negative integers (p, q) such that (p, q) not equal (1, 1) and p q = k - 3. This resolves an open problem of Bonamy et al. (2017). Combined with results of Borodin et al. (2000), Yang and Yuan (2006) and Wu et al. (1996), it also settles the complexity of deciding whether a graph with bounded maximum degree can be partitioned into two subgraphs of prescribed degeneracy. (C) 2019 The Author(s). Published by Elsevier Ltd.