AbstractLet G be a graph with maximum degree d≥ 3 and ω(G)≤ d, where ω(G) is the clique number of the graph G. Let p1 and p2 be two positive integers such that d = p1 + p2. In this work, we prove that G has a vertex partition S1, S2 such that G[S1] is a maximum order (p1‐1)‐degenerate subgraph of G and G[S2] is a (p2‐1)‐degenerate subgraph, where G[Si] denotes the graph induced by the set Si in G, for i = 1,2. On one hand, by using a degree‐equilibrating process our result implies a result of Bollobas and Marvel [1]: for every graph G of maximum degree d≥ 3 and ω(G)≤ d, and for every p1 and p2 positive integers such that d = p1 + p2, the graph G has a partition S1,S2 such that for i = 1,2, Δ(G[Si])≤ pi and G[Si] is (pi‐1)‐degenerate. On the other hand, our result refines the following result of Catlin in [2]: every graph G of maximum degree d≥ 3 has a partition S1,S2 such that S1 is a maximum independent set and ω(G[S2])≤ d‐1; it also refines a result of Catlin and Lai [3]: every graph G of maximum degree d≥ 3 has a partition S1,S2 such that S1 is a maximum size set with G[S1] acyclic and ω(G[S2])≤ d‐2. The cases d = 3, (d,p1) = (4,1) and (d,p1) = (4,2) were proved by Catlin and Lai [3]. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 227–232, 2007