The chromatic number of a graph is the smallest number of colors needed to color its vertices so that no two adjacent vertices share the same color. This concept is important in graph theory, a branch of mathematics that studies the properties of graphs, which are structures made up of vertices (or nodes) connected by edges. Determining the chromatic number can help solve various problems, such as scheduling tasks or coloring maps. For example, the famous Four Color Theorem states that any map can be colored with no more than four colors without adjacent regions sharing the same color, illustrating a practical application of chromatic numbers.