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.