Weak acyclic coloring and asymmetric coloring games

Abstract

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 ) ⩽ 1 2 ( k 2 + 3 k ) .