The graph coloring problem is a way to assign colors to the vertices of a graph so that no two adjacent vertices share the same color. This is useful in various applications, such as scheduling tasks or assigning frequencies in wireless networks. The goal is to minimize the number of colors used, which is known as the graph's chromatic number. This problem can be represented mathematically and is a classic example in the field of combinatorial optimization. It has practical implications in areas like map coloring, where different regions must be colored differently to avoid confusion, and in register allocation in computer programming, where variables need to be assigned to limited resources efficiently.