Weak acyclic coloring and asymmetric coloring games

We introduce the notion of weak acyclic coloring of a graph. This is a relaxation of the usual notion of acyclic coloring which is often sufficient for applications. We then use this concept to analyze the ( a, b )-coloring game. This game is played on a finite graph G, using a set of colors X, by two players Alice and Bob with Alice playing first. On each turn Alice (Bob) chooses a (b) uncolored vertices and properly colors them with colors from X. Alice wins if the players eventually create a proper coloring of G; otherwise Bob wins when one of the players has no legal move. The ( a, b )-game chromatic number of G, denoted ( a, b )- χ_g ( G ), is the least integer t such that Alice has a winning strategy when the game is played on G using t colors. We show that if the weak acyclic chromatic number of G is at most k then ( 2, 1 )- χ_g ( G ) {less-than or slanted equal to} frac(1, 2) ( k² + 3 k ).