Asymmetric directed graph coloring games

This note generalizes the (a,b)(a,b)-coloring game and the (a,b)(a,b)-marking game which were introduced by Kierstead [H.A. Kierstead, Asymmetric graph coloring games, J. Graph Theory 48 (2005) 169–185] for undirected graphs to directed graphs. We prove that the (a,b)(a,b)-chromatic and (a,b)(a,b)-coloring number for the class of orientations of forests is b+2b+2 if b≤ab≤a, and infinity otherwise. From these results we deduce upper bounds for the (a,b)(a,b)-coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth.