Coloring games on squares of graphs

Abstract

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)=max{2|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.