Graph colorings are ubiquitous in the modeling of real-world problems. There are many applications and conjectures, which are still open and studied by various mathematicians and computer scientists. In this paper, we deal with cost graph colorings as an important subfield of graph colorings. In cost graph coloring, each color has a distinct cost, and we need to pay the cost each time to color each vertex or edge. Our task is to find a coloring with the minimum total cost. The cost coloring problems are NP-hard in general; however, polynomial time algorithms are known for certain classes of graphs.