Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph GG. An asymmetric variant of the (r,d)(r,d)-relaxed coloring game is called the (r,d)(r,d)-relaxed (a,b)(a,b)-coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph GG, using colors from a set XX, with |X|=r|X|=r. On each turn Alice colors aa vertices and Bob colors bb vertices. A color α∈Xα∈X is legal for an uncolored vertex uu if by coloring uu with color αα, the subgraph induced by all the vertices colored with αα has maximum degree at most dd. Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The dd-relaxed (a,b)(a,b)-game chromatic number, denoted by (a,b)-χgd(G), of GG is the least rr for which Alice has a winning strategy in the (r,d)(r,d)-relaxed (a,b)(a,b)-coloring game.The (r,d)(r,d)-relaxed (1,1)(1,1)-coloring game has been well studied and there are many interesting results. For the (r,d)(r,d)-relaxed (a,1)(a,1)-coloring game, this paper proves that if a graph GG has an orientation with maximum outdegree kk and a≥ka≥k, then (a,1)-χgd(G)≤k+1 for all d≥k2+2kd≥k2+2k; If a≥k3a≥k3, then (a,1)(a,1)-χgd(G)≤k+1 for all d≥2k+1d≥2k+1.