On the Hardness of Approximating the Chromatic Number


-colorable for some fixed . 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 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 for some fixed ε > 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 , such that it is NP-hard to decide whether a given graph G is -chromatic or any coloring of G requires colors.

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Received April 11, 1997/Revised June 10, 1999

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Khanna, S., Linial, N. & Safra, S. On the Hardness of Approximating the Chromatic Number. Combinatorica 20, 393–415 (2000). https://doi.org/10.1007/s004930070013

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  • AMS Subject Classification (1991) Classes:  68Q17, 68Q25, 68R10