On the hardness of approximating the chromatic number

We study the hardness of approximating the chromatic number when the input graph is k-colorable for some fixed k≥3. Our main result is that it is NP-hard to find a 4-coloring of a 3-chromatic graph. As an immediate corollary we obtain that it is NP-hard to color a k-chromatic graph with at most k+2⌊k/3⌋ - 1 colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NP-hard to approximate the chromatic number to within n^∈ for some fixed ∈ > 0. We point here that this hardness result applies only to graphs with large chromatic numbers. The second result shows that for any positive constant h, there exists an integer k_h, such that it is NP-hard to decide whether a given graph G is k_h-chromatic or any coloring of G requires h·k_h colors.