Generalised game colouring of graphs Original Research Article

We consider the version of a colouring game introduced by Bodlaender [On the complexity of some colorings games, Internat. J. Found. Comput. Sci. 2 (1991) 133–147]. We combine the concepts: this game and the generalised colouring of graphs as follows. The two players are Alice and Bob and they play alternatively with Alice having the first move. Let be given a graph G and an ordered set of hereditary properties image. The players take turns colouring G with colours from image such that for each image the induced subgraph image (image is the set of vertices of G with colour i) has the property image after each move of the players. If after image moves the graph G is image-partitioned (generalised coloured) then Alice wins. In this case, we say that the graph G has the property image. We characterise the class image of graphs and we give an answer to a question, for image, posed by Zhu [The game coloring number of planar graphs, J. Combin. Theory B 75 (1999) 245–258]. We describe a new strategy for Alice for playing the image-game on acyclic graphs. Also some open problems are posed.