On the hardness of approximating the chromatic number

The paper considers the computational hardness of approximating the chromatic number of a graph. The authors first give a simple proof that approximating the chromatic number of a graph to within a constant power (of the value itself) in NP-hard. They then consider the hardness of coloring a 3-colorable graph with as few as possible colors. They show that determining whether a graph is 3-colorable or any legal coloring of it requires at least 5 colors is NP-hard. Therefore, coloring a 3-colorable graph with 4 colors is NP-hard