Adapted game colouring of graphs

Suppose G=(V,E)G=(V,E) is a graph and FF is a colouring of its edges (not necessarily proper) that uses the colour set XX. In an adapted colouring game, Alice and Bob alternately colour uncoloured vertices of GG with colours from XX. A partial colouring cc of the vertices of GG is legal   if there is no edge e=xye=xy such that c(x)=c(y)=F(e)c(x)=c(y)=F(e). In the process of the game, each partial colouring must be legal. If eventually all the vertices of GG are coloured, then Alice wins the game. Otherwise, Bob wins the game. The adapted game chromatic number   of a graph GG, denoted by χadg(G), is the minimum number of colours needed to ensure that for any edge colouring FF of GG, Alice has a winning strategy for the game. This paper studies the adapted game chromatic number of various classes of graphs. We prove that the maximum adapted game chromatic number of trees is 33, the maximum adapted game chromatic number of outerplanar graphs is 55, the adapted game chromatic number of partial kk-trees is at most 2k+12k+1, and the adapted game chromatic number of planar graphs is at most 1111.