The game coloring number of the square of a graph G, denoted by gcol( G2), was first studied by Esperet and Zhu. The (a,b)-game coloring number, denoted by (a,b)-gcol(G), is defined like the game coloring number, except that on each turn Alice makes a moves and Bob makes b moves. For a graph G, the maximum average degree of G is defined as Mad(G)=max2|E(H)||V(H)|:H is a subgraph of G. Let k be an integer. In this paper, by introducing a new parameter rG, which is defined through orientations and orderings of the vertices of G, we show that if a<Mad(G)2≤k, then (a,1)-gcol( G2)≤kΔ(G)+⌊(1+1a) rG⌋+ rG+2. This implies that if G is a partial k-tree and a<k, then (a,1)-gcol( G2)≤kΔ(G)+(1+1a)( k2+3k+22)+2; if G is planar, then there exists a constant C such that gcol( G2) ≤5Δ(G)+C. These improve previous corresponding known results. For a<k<Mad(G) and Δ(G)<2k-2, we prove that (a,1)-gcol( G2)≤(3k-2)Δ(G)- k2+4k+2. © 2012 Elsevier B.V. All rights reserved.
Yang, D. (2012). Coloring games on squares of graphs. Discrete Mathematics, 312(8), 1400–1406. https://doi.org/10.1016/j.disc.2012.01.004