Decomposition of sparse graphs, with application to game coloring number

Let kk be a nonnegative integer, and let mk=4(k+1)(k+3)k2+6k+6. We prove that every simple graph with maximum average degree less than mkmk decomposes into a forest and a subgraph with maximum degree at most kk (furthermore, when k≤3k≤3 both subgraphs can be required to be forests). It follows that every simple graph with maximum average degree less than mkmk has game coloring number at most 4+k4+k.