The difference between game chromatic number and chromatic number of graphs

Abstract

The graph coloring game on a graph G is a two-player game. In the game, the players alternately color an uncolored vertex of G by a color in a given color set X so that any two adjacent vertices receive different colors. The first player's aim is to completely color all vertices of G only by using colors in X, and the second player's aim is to avoid it. The game chromatic number of a graph G, denoted by χg(G), is the minimum number of colors such that the first player has a winning strategy for the graph coloring game on G. In this paper, we prove that for any simple graph G, χg(G)−χ(G)≤⌊n2⌋−1, where χ(G) is the chromatic number of G. Moreover, the estimation is best possible since there exist graphs with even order attaining the equality.