Asymmetric graph coloring games

We introduce the (a, b)-coloring game, an asymmetric version of the coloring game played by two players Alice and Bob on a finite graph, which differs from the standard version in that, in each turn, Alice colors a vertices and Bob colors b vertices. We also introduce a related game, the (a, b)-marking game. We analyze these games and determine the (a, b)chromatic numbers and (a, b)-coloring numbers for the class of forests and all Values of a and b.