Introduction:
In graph coloring problems, we are asked to color the parts of the graph.
Vertex coloring is one of the most common graph coloring problems. In this problem, we are given a graph and ‘m’ colors. We need to find a number of ways to color a graph with these m colors such that no two adjacent nodes are colored the same. Some of the other graph coloring problems, like Edge Coloring (No vertex is incident to two edges of the same color) and Face Coloring (Geographical Map Coloring), can be transformed into vertex coloring.
Another famous graph coloring problem is Chromatic Number. In this problem, we need to find the minimum number of colors are required to color the graph such that no adjacent nodes are colored the same.
In the above graph, we can see that we need 3 colors to color the entire graph such that no two nodes are colored the same.
Applications of Graph Coloring:

Sudoku:
 This problem can also be considered a graph coloring problem. We can connect two cells with an edge if they are in the same row or same column and then color each node with no adjacent nodes having the same color.

Bipartite Graph:
 A graph can be checked if it is a bipartite graph or not by checking if we can color a graph with two colors such that no adjacent nodes have the same color.

Making Schedule or Time Table:
 Suppose we want to make an exam schedule for a university. We have listed different subjects and students enrolled in every subject. Many subjects would have common students (of the same batch, some backlog students, etc). How do we schedule an exam so that no two exams with a common student are scheduled at the same time? How many minimum different time slots are needed to schedule all exams? This problem can be solved using a graph where every vertex is a subject, and an edge between two vertices means there is a common student. Hence, it is a graph coloring problem where a minimum number of time slots is equal to the chromatic number of the graph.

Register Allocation:
 In compiler optimization, Register allocation refers to assigning variables to registers and handling the transfer of data into and out of registers. This is also a graph coloring problem.

Frequency allocation:
 The mobile towers of the same location should be assigned a different frequency. Hence, this can be solved using the graph coloring problem where towers are assumed to be nodes and two towers are connected with an edge if they are in the same location.