Lightness of digraphs in surfaces and directed game chromatic number

The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the in-degrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum in-degree at least 1 (resp., graphs with minimum degree at least 2) and a given girth  k , and without 4-cycles, which can be embedded in a surface  S . (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k ≥ 5 , these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [W. He, X. Hou, K.-W. Lih, J. Shao, W. Wang, X. Zhu, Edge-partitions of planar graphs and their game coloring numbers, J. Graph Theory 41 (2002) 307–317] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game colouring number, and thus for the game chromatic number of (di)graphs with girth  k and without 4-cycles embeddable in  S . The game chromatic resp. game colouring number were introduced by Bodlaender [H.L. Bodlaender, On the complexity of some coloring games, Int. J. Found. Comput. Sci. 2 (1991) 133–147] resp. Zhu [X. Zhu, The game coloring number of planar graphs, J. Combin. Theory B 75 (1999) 245–258] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game colouring number of a directed simple forest is at most 3.