The game of arboricity

Using a fixed set of colors C, Ann and Ben color the edges of a graph G so that no monochromatic cycle may appear. Ann wins if all edges of G have been colored, while Ben wins if completing a coloring is not possible. The minimum size of C for which Ann has a winning strategy is called the game arboricity of G  , denoted by Ag(G)Ag(G). We prove that Ag(G)⩽3kAg(G)⩽3k for any graph G of arboricity k  , and that there are graphs such that Ag(G)⩾2k-2Ag(G)⩾2k-2. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide two other strategies based on induction and acyclic colorings.