Geodesic
Note On The Game Colouring Number Of Powers Of Graphs
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 31-42

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We generalize the methods of Esperet and Zhu [6] providing an upper bound for the game colouring number of squares of graphs to obtain upper bounds for the game colouring number of m-th powers of graphs, m ≥ 3, which rely on the maximum degree and the game colouring number of the underlying graph. Furthermore, we improve these bounds in case the underlying graph is a forest.
Keywords: game colouring number, marking game, graph power, game chromatic number, forest 
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Andres, Stephan Dominique; Theuser, Andrea. Note On The Game Colouring Number Of Powers Of Graphs. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 1, pp. 31-42. http://geodesic.mathdoc.fr/item/DMGT_2016_36_1_a2/