Consider a graph G=(V,E) of order n. In the minimum graph-coloring problem we try to color V with as few colors as possible so that no two adjacent vertices receive the same color. This problem is among the first ones proved to be intractable, and hence, it is very unlikely that an optimal polynomial-time algorithm could ever be devised for it. In this paper, we survey the main polynomial time approximation algorithms (the ones for which theoretical approximability bounds have been studied) for the minimum graph-coloring and we discuss their approximation performance and their complexity. Finally, we further improve the approximation ratio for graph-coloring.
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