Competitive Colorings of Oriented Graphs

Nešeťril and Sopena introduced a concept of oriented game chromatic number and developed a general technique for bounding this parameter. In this paper, we combine their technique with concepts introduced by several authors in a series of papers on game chromatic number to show that for every positive integer k, there exists an integer t so that if C is a topologically closed class of graphs and C does not contain a complete graph on k vertices, then whenever G is an orientation of a graph from C, the oriented game chromatic number of G is at most t. In particular, oriented planar graphs have bounded oriented game chromatic number. This answers a question raised by Nešeťril and Sopena. We also answer a second question raised by Nešeťril and Sopena by constructing a family of oriented graphs for which oriented game chromatic number is bounded but extended Go number is not.