Coloring games on squares of graphs

The game coloring number of the square of a graph GG, denoted by gcol(G2)gcol(G2), was first studied by Esperet and Zhu. The (a,b)(a,b)-game coloring number, denoted by (a,b)(a,b)-gcol(G)gcol(G), is defined like the game coloring number, except that on each turn Alice makes aa moves and Bob makes bb moves. For a graph GG, the maximum average degree of GG is defined as Mad(G)=max{2|E(H)||V(H)|:H is a subgraph of G}G}. Let kk be an integer. In this paper, by introducing a new parameter rGrG, which is defined through orientations and orderings of the vertices of GG, we show that if a