Decomposition of sparse graphs, with application to game coloring number

Let k be a nonnegative integer, and let mk = 4(k+1)(k+3) k2. We prove that every +6k+6 simple graph with maximum average degree less than mk decomposes into a forest and a subgraph with maximum degree at most k (furthermore, when k ≤ 3 both subgraphs can be required to be forests). It follows that every simple graph with maximum average degree less than mk has game coloring number at most 4 + k.