Game Chromatic Number of Some Regular Graphs

Let G be a graph and k be a positive integer. We consider a game with two players Alice and Bob who alternate in coloring the vertices of G with a set of k colors. In every turn, one vertex will be chosen by one player. Alice’s goal is to color all vertices with the k colors, while Bob’s goal is to prevent her. The game chromatic number denoted by χg(G), is the smallest k such that Alice has a winning strategy with k colors. In this paper, we determine the game chromatic number χg of circulant graphs Cn(1,2), , and generalized Petersen graphs GP(n,2), GP(n,3).