Decomposition of sparse graphs, with application to game coloring number

Let k be a nonnegative integer, and let m k = 4 ( k + 1 ) ( k + 3 ) k 2 + 6 k + 6 . We prove that every simple graph with maximum average degree less than m k 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 m k has game coloring number at most 4 + k .