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Coloring Random Graphs — an Algorithmic Perspective

  • Michael Krivelevich
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Algorithmic Graph Coloring and Random Graphs have long become one of the most prominent branches of Combinatorics and Combinatorial Optimization. It is thus very natural to expect that their mixture will produce quite many very attractive, diverse and challenging problems. And indeed, the last thirty or so years witnessed rapid growth of the field of Algorithmic Random Graph Coloring, with many researchers working in this area and bringing there their experience from different directions of Combinatorics, Probability and Computer Science. One of the most distinctive features of this field is indeed the diversity of tools and approaches used to tackle its central problems.

Keywords

Greedy Algorithm Random Graph Chromatic Number Graph Coloring Common Neighbor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    N. Alon and N. Kahale A spectral technique for coloring random 3-colorable graphs Proc. of the 26thAnnual ACM Symposium on Theory of Computing (STOC’94), 346–355.Google Scholar
  2. [2]
    N. Alon and M. Krivelevich The concentration of the chromatic number of random graphs Combinatorica 17 (1997), 303–313.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    N. Alon, M. Krivelevich and B. Sudakov List coloring of random and pseudorandom graphs Combinatorica 19 (1999), 453–472.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    N. Alon, M. Krivelevich and V. H. Vu On the concentration of eigenvalues of random symmetric matrices Israel Journal of Mathematics, in press.Google Scholar
  5. [5]
    N. Alon and J. Spencer, The probabilistic method, 2“ ed., Wiley, New York, 2000.CrossRefGoogle Scholar
  6. [6]
    E. Bender and H. Wilf A theoretical analysis of backtracking in the graph coloring problem J. Algorithms 6 (1985), 275–282.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A. Blum and J. Spencer Coloring random and semirandom k-colorable graphs J. Algorithms 19 (1995), 204–234.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    B. Bollobás The chromatic number of random graphs Combinatorica 8 (1988), 49–55.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    B. Bollobás, Random graphs, 2nded., Cambridge Univ. Press, Cambridge, 2001.zbMATHCrossRefGoogle Scholar
  10. [10]
    J. Culberson and I. Gent Well out of reach: why hard problems are hard Technical report APES-13–1999, APES Research Group, 1999. Available from:http://www.cs.strath.ac.uk/apes/apesreports.html.
  11. [11]
    M. Dyer and A. Frieze The solution of some random NP-hard problems in polynomial expected time J. Algorithms 10 (1989), 451–489.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    U. Feige and J. Kilian Zero knowledge and the chromatic number Proc. 11th IEEE Conf. Comput. Complexity, IEEE (1996), 278–287.Google Scholar
  13. [13]
    U. Feige and J. Kilian Heuristics for semirandom graph problems J. Computer and System Sciences, to appear.Google Scholar
  14. [14]
    Z. Füredi and J. Komlós The eigenvalues of random symmetric matrices Combinatorica 1 (1981), 233–241.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    M. Fürer, C. R. Subramanian and C. E. Veni Madhavan Coloring random graphs in polynomial expected time Algorithms and Comput. (Hong Kong 1993), Lecture Notes Comp. Sci. 762, Springer, Berlin, 1993, 31–37.Google Scholar
  16. [16]
    G. Grimmett and C. McDiarmid On colouring random graphs Math. Proc. Cam. Phil. Soc. 77 (1975), 313–324.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    M. Grötschel, L. Lovász and A. Schrijver, Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics 2, Springer Verlag, Berlin, 1993.CrossRefGoogle Scholar
  18. [18]
    V. Guruswami and S. Khanna On the hardness of 4-coloring a 3-colorable graph Proc. 15thAnnual IEEE Conf. on Computational Complexity, IEEE Comput. Soc. Press, Los Alamitos, 2000, 188–197.Google Scholar
  19. [19]
    M. M. Halldórsson, A still better performance guarantee for approximate graph coloring Inform. Process. Letters 45 (1993), 19–23.zbMATHGoogle Scholar
  20. [20]
    S. Janson, T. Luczak and A. Ruciáski, Random graphs, Wiley, New York, 2000.zbMATHCrossRefGoogle Scholar
  21. [21]
    T. Jensen and B. Toft, Graph coloring problems, Wiley, New York, 1995.zbMATHGoogle Scholar
  22. [22]
    M. Jerrum Large cliques elude the metropolis process Random Structures and Algorithms 3 (1992), 347–359.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    A. Juels and M. Peinado, Hiding cliques for cryptographic security Proc. of the Ninth Annual ACM-SIAM SODA ACM Press (1998), 678–684.Google Scholar
  24. [24]
    D. Karger and R. Motwani and M. Sudan Approximate Graph coloring by semidefinite programming Journal of the ACM, 45 (1998), 246–265.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    R. Karp, Reducibility among combinatorial problems, in: Complexity of computer computations (E. Miller and J. W. Thatcher, eds.) Plenum Press, New York, 1972, 85–103.CrossRefGoogle Scholar
  26. [26]
    R. M. Karp, Probabilistic analysis of some combinatorial search problems, In: Algorithms and Complexity: New Directions and Recent Results J. F. Traub, ed., Academic Press, New York, 1976, pp. 1–19.Google Scholar
  27. [27]
    H. Kierstead Coloring graphs on-line Online algorithms (Schloss-Dagstuhl 1996), Lecture Notes in Computer Science 1442, Springer, Berlin, 1998,281– 305.Google Scholar
  28. [28]
    S. Khanna, N. Linial and S. Safra On the hardness of approximating the chromatic number Combinatorica 20 (2000), 393–415.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    M. Krivelevich The choice number of dense random graphs Combinatorics, Probability and Computing 9 (2000), 19–26.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    M. Krivelevich Deciding k-colorability in expected polynomial time Information Processing Letters 81 (2002), 1–6.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    M. Krivelevich and B. Sudakov Coloring random graphs Inform. Process. Letters 67 (1998), 71–74.MathSciNetGoogle Scholar
  32. [32]
    M. Krivelevich and V. H. Vu Approximating the independence number and the chromatic number in expected polynomial time Proc. 27thInt. Colloq. on Automata, Languages and Programming (ICALP’2000), Lecture Notes in Computer Science 1853, Springer, Berlin, 13–24.Google Scholar
  33. [33]
    M. Krivelevich and V. H. Vu Choosability in random hypergraphs Journal of Combinatorial Theory Ser. B 83 (2001), 241–257.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    L. Kucera Graphs with small chromatic numbers are easy to color Inform. Process. Letters 30 (1989), 233–236.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    L. Kucera The greedy coloring is a bad probabilistic algorithm J. Algorithms 12 (1991), 674–684.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    L. Lovász On the Shannon capacity of a graph IEEE Trans. Inform. Theory 25 (1979), 1–7.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    T. Luczak The chromatic number of random graphs Combinatorica 11 (1991), 45–54.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    T. Luczak A note on the sharp concentration of the chromatic number of random graphs Combinatorica 11 (1991), 295–297.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    C. J. H. McDiarmid Colouring random graphs badly Graph Theory and Combinatorics (R. J. Wilson, Ed.), Pitman Research Notes in Mathematics, vol. 34, 76–86.Google Scholar
  40. [40]
    F. McSherry Spectral partitioning of random graphs Proc. 42ndIEEE Symposium on Found. of Comp. Science (FOCS’01), IEEE Comp. Society, 529–537.Google Scholar
  41. [41]
    D. W. Matula On the complete subgraph of a random graph Combinatory mathematics and its applications, Chapel Hill, North Carolina (1970), 356–369.Google Scholar
  42. [42]
    D. Matula Expose-and-merge exploration and the chromatic number of a random graph Combinatorica 7 (1987), 275–284.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    M. Molloy Thresholds for colourability and satisfiability in random graphs and Boolean formulae in: Surveys in Combinatorics 2001, London Math. Soc. Lect. Note Ser. 288, Cambridge Univ. Press, Cambridge, 2001.Google Scholar
  44. [44]
    B. Pittel and R. S. Weishaar On-line coloring of sparse random graphs and random trees J. Algorithms 23 (1997), 195–205.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [45]
    E. Shamir and J. Spencer Sharp concentration of the chromatic number of random graphs G n p Combinatorica 7 (1987), 124–129.MathSciNetCrossRefGoogle Scholar
  46. [46]
    E. Shamir and E. Upfal Sequential and distributed graph coloring algorithms with performance analysis in random graph spaces J. Algorithms 5 (1984), 488–501.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    C. R. Subramanian Minimum coloring k-colorable graphs in polynomial average time J. Algorithms 33 (1999), 112–123.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    C. R. Subramanian Algorithms for colouring random k-colourable graphs Combinatorics, Probability and Computing 9 (2000), 45–77.zbMATHCrossRefGoogle Scholar
  49. [49]
    C. R. Subramanian, M. Fürer and C. E. Veni Madhavan Algorithms for coloring semi-random graphs Random Struct. Algorithms 13 (1998), 125–158.zbMATHCrossRefGoogle Scholar
  50. [50]
    J. S. Turner Almost all k-colorable graphs are easy to color J. Algorithms 9 (1988), 63–82.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    J. H. van Lint and R. M. Wilson, A course in combinatorics, Cambridge Univ. Press, Cambridge, 1992.Google Scholar
  52. [52]
    H. Wilf Backtrack: a 0(1) expected time algorithm for the graph coloring problem Inform. Proc. Letters 18 (1984), 119–121.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Michael Krivelevich
    • 1
  1. 1.Department of Mathematics Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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