Combinatorica

, Volume 20, Issue 3, pp 393–415 | Cite as

On the Hardness of Approximating the Chromatic Number

  • Sanjeev Khanna
  • Nathan Linial
  • Shmuel Safra
Original Paper

k

-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.

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

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Copyright information

© János Bolyai Mathematical Society, 2000

Authors and Affiliations

  • Sanjeev Khanna
    • 1
  • Nathan Linial
    • 2
  • Shmuel Safra
    • 3
  1. 1.Department of Computer and Information Science, University of Pennsylvania; Philadelphia, PA 19104, USA; E-mail: sanjeev@cis.upenn.eduUS
  2. 2.Institute of Computer Science, Hebrew University of Jerusalem; Jerusalem 91904, Israel; E-mail: nati@cs.huji.ac.ilUS
  3. 3.Department of Computer Science, Tel Aviv University; Tel Aviv 69978, Israel; E-mail: safra@math.tau.ac.ilIL

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