Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph G . An asymmetric variant of the ( r , d ) -relaxed coloring game is called the ( r , d ) -relaxed ( a , b ) -coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph G , using colors from a set X , with | X | = r . On each turn Alice colors a vertices and Bob colors b vertices. A color α ∈ X is legal for an uncolored vertex u if by coloring u with color α , the subgraph induced by all the vertices colored with α has maximum degree at most d . 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 d -relaxed ( a , b ) -game chromatic number, denoted by ( a , b ) - χ g d ( G ) , of G is the least r for which Alice has a winning strategy in the ( r , d ) -relaxed ( a , b ) -coloring game. r , d ) -relaxed ( 1 , 1 ) -coloring game has been well studied and there are many interesting results. For the ( r , d ) -relaxed ( a , 1 ) -coloring game, this paper proves that if a graph G has an orientation with maximum outdegree k and a ≥ k , then ( a , 1 ) - χ g d ( G ) ≤ k + 1 for all d ≥ k 2 + 2 k ; If a ≥ k 3 , then ( a , 1 ) - χ g d ( G ) ≤ k + 1 for all d ≥ 2 k + 1 .