An approximation algorithm for 3-Colourability

  • Ingo Schiermeyer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1017)


We present a polynomial time approximation algorithm to colour a 3-colourable graph G with 3f(n) colours, if G has minimum degree δ(G)≥αn/f(n), where Ω(1)f(n)<O(n) and α is a positive constant. We also discuss NP—completeness and #P—completeness of restricted k-Colourability problems.

Key words

Graph k-colouring exact and approximation algorithm complexity 


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  1. [1]
    A. Blum, New Approximation Algorithms for Graph Colouring, J. ACM 41 (1994) 470–516.Google Scholar
  2. [2]
    J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).Google Scholar
  3. [3]
    N. Christofides, An Algorithm for the Chromatic Number of a Graph, Computer J. 14 (1971) 38–39.Google Scholar
  4. [4]
    K. Edwards, The Complexity of Colouring Problems on Dense Graphs, Theoretical Computer Sciene 43 (1986) 337–343.Google Scholar
  5. [5]
    S. Even, A. Itai and A. Shamir, On the complexity of timetable and multicommodity flow problems, SIAM J. Comput. 5 (1976) 691–703.CrossRefGoogle Scholar
  6. [6]
    M. R. Garey and D. S. Johnson, The complexity of Near-Optimal Graph Coloring, J. ACM 23 (1976) 43–49.Google Scholar
  7. [7]
    M. R. Garey and D. S. Johnson, Computers and Intractability, A Guide to the Theory of N P-Completeness, W. H. Freeman and Company, San Francisco, 1979.Google Scholar
  8. [8]
    S. Khanna, N. Linial and S. Safra, On the hardness of approximating the chromatic number, Proc. of the 2nd Israel Symp. on the Theory of Computing and Systems, IEEE Computer Society, 250–260.Google Scholar
  9. [9]
    E. L. Lawler, A Note on the Complexity of the Chromatic Number Problem, Inform. Process. Lett. 5 (1976) 66–67.Google Scholar
  10. [10]
    L. Lovász, On the Ratio of Optimal and Fractional Covers, Discrete Math. 13 (1975) 383–390.CrossRefGoogle Scholar
  11. [11]
    C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, Proc. of the Annual ACM Symposium on the Theory of Computing, Vol.25 (1993) 286–293.Google Scholar
  12. [12]
    B. Monien and E. Speckenmeyer, Solving Satisfiability in less than 2n Steps, Discrete Appl. Math. 10 (1985) 287–295.CrossRefGoogle Scholar
  13. [13]
    J. W. Moon and L. Moser, On Cliques in Graphs, Israel J. of Math. 3 (1965) 23–28.Google Scholar
  14. [14]
    J. M. Robson, Algorithms for Maximum Independent Sets, J. of Alg. 7 (1986) 425–440.Google Scholar
  15. [15]
    I. Schiermeyer, Solving 3-Satisfiability in less than 1,579n Steps, Lecture Notes in Computer Science 702 (1993) 379–394.Google Scholar
  16. [16]
    I. Schiermeyer, Fast exact Colouring Algorithms, RWTH Aachen, preprint 1993, Journal Tatra Mountains Mathematical Publications, in print.Google Scholar
  17. [17]
    R. E. Tarjan and A. E. Trojanowski, Finding a Maximum Independent Set, SIAM J. Comput., Vol. 6, No. 3, September 1977, 537–546.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Ingo Schiermeyer
    • 1
  1. 1.Lehrstuhl für Diskrete Mathematik und Grundlagen der InformatikTechnische UniversitÄt CottbusCottbusGermany

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