Asymmetric directed graph coloring games

This note generalizes the (a, b)-coloring game and the (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)chromatic and (a, b)-coloring number for the class of orientations of forests is b + 2 if b <= a, and infinity otherwise. From these results we deduce upper bounds for the (a, b)coloring number of oriented outerplanar graphs and of orientations of graphs embeddable in a surface with bounded girth. (C) 2008 Elsevier B.V. All rights reserved.