Colorability Saturation Games

We consider the following two-player game: Maxi and Mini start with the empty graph on n vertices and take turns, always adding one additional edge to the graph such that the chromatic number is at most k, where k ∈ N is a given parameter. The game is over when the graph is saturated and no further edge can be inserted. Maxi wants to maximize the length of the game while Mini wants to minimize it. The score s(n, k) denotes the final number of edges in the graph, given that both players played optimally. This colorability game belongs to the general class of saturation games which aroused interest in the field of combinatorial games during the last years, as they exhibit simple yet challenging problems. The analysis of the described colorability saturation game has been initiated recently by Hefetz et al. [European J. of Comb., Vol. 51(C), 2016]. In this paper, we provide almost-matching lower and upper bounds on the score s(n, k) of the game. In addition, we consider the special case k = 4 for which we prove s(n, 4) = n/3 +O(n).