A New Game Chromatic Number

Consider the following two-person game on a graphG.Players I and II move alternatively to color a yet uncolored vertex ofGproperly using a pre-specified set of colors. Furthermore, Player II can only use the colors that have been used, unless he is forced to use a new color to guarantee that the graph is colored properly. The game ends when some player can no longer move. Player I wins if all vertices ofGare colored. Otherwise Player II wins. What is the minimal numberχg*(G)of colors such that Player I has a winning strategy? This problem is motivated by the game chromatic numberχg(G)introduced by Bodlaender and by the continued work of Faigle, Kern, Kierstead and Trotter. In this paper, we show thatχg*(T) ≤3for each treeT.We are also interested in determining the graphsGfor whichχ(G)= χg*(G),as well asχg*(G)for thek-inductive graphs wherekis a fixed positive integer.