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 . 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 kh, such that it is NP-hard to decide whether a given graph G is kh-chromatic or any coloring of G requires h·kh colors.