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)(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)(a,b)-game chromatic number of G  , denoted (a,b)(a,b)-χg(G)χg(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)(2,1)-χg(G)⩽12(k2+3k).