Game chromatic number of graphs

he game chromatic number of a graph $G$ was first defined by Bodlaender using a two-player coloring game. Let $G$ be a finite graph and $X$ a set of colors. Two players, commonly referred to as Alice and Bob, take turns coloring the vertices of the graph using colors from $X$, with Alice starting first. The goal is to ensure that adjacent vertices are colored differently. If all vertices of the graph can be colored in this way, Alice wins the game. However, if at any point an uncolored vertex remains that is adjacent to vertices already colored with all colors in $X$, Bob wins the game. The game chromatic number of a graph $G$, denoted by $\chi_g(G)$, is defined as the minimum number of colors in $X$ such that Alice can always win with an optimal strategy. In this study, the game chromatic numbers of various graph families and Cartesian products of specific graphs are investigated, and results obtained from existing research are compiled and presented.