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Graph coloring is a fundamental problem in graph theory. It involves assigning labels, typically called colors, to the vertices of a graph such that no two adjacent vertices share the same color. The objective is usually to minimize the number of colors used, which is known as the chromatic number of the graph.

The concept of graph coloring has numerous applications across various fields. One common application is in scheduling problems, such as exam scheduling or course timetabling. In this context, courses or exams are represented as vertices, and an edge exists between two vertices if the corresponding courses have a student in common or clash in some way. Assigning colors to vertices then corresponds to assigning time slots, where adjacent vertices must have different time slots.

Another significant application is in register allocation in computer compilers. Here, variables are represented as vertices, and an edge exists between two variables if they are "live" at the same time, meaning their values might be needed simultaneously. The colors represent the available registers, and the coloring problem aims to assign variables to registers efficiently to minimize the need for temporary storage.

Map coloring is a classic example, where regions on a map are vertices and adjacent regions are connected by edges. The problem is to color the regions such that no two adjacent regions have the same color, using the minimum number of colors. The Four Color Theorem states that any planar graph can be colored with at most four colors.

The problem of determining the chromatic number of a graph is NP-hard in general, meaning that for large graphs, finding the absolute minimum number of colors can be computationally very expensive. Consequently, many algorithms focus on finding approximate solutions or heuristics that provide a good coloring, even if it's not strictly optimal.