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Colouring random graphs

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Abstract

We discuss some results concerned with the behaviour of colouring algorithms on large random graphs.

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References

  1. E.A. Bender and H.S. Wilf, A theoretical analysis of backtracking in the graph coloring problem (1983) preprint.

  2. B. Bollobás, Graph Theory, Graduate Texts in Mathematics 63 (Springer Verlag, Berlin, 1979).

    Google Scholar 

  3. B. Bollobás and P. Erdős, Cliques in random graphs, Math. Proc. Camb. Phil. Soc. 80 (1976) 419.

    Google Scholar 

  4. D. Brélaz, New methods to color the vertices of a graph, Comm. A.C.M. 22 (1979) 251.

    Google Scholar 

  5. J.R. Brown, Chromatic scheduling and the chromatic number problem, Man. Sci. 19 (1972) 456.

    Google Scholar 

  6. Y.D. Burtin, Aymptotic estimates of the diameter, independence and dominance numbers of a random graph, Sov. Math. Dokl. 14 (1973) 497.

    Google Scholar 

  7. N. Christofides, An algorithm for the chromatic number of a graph, Comput. J. 14 (1971) 38.

    Google Scholar 

  8. N. Christofides, Graph Theory — an Algorithmic Approach (Academic Press, 1975).

  9. V. Chvátal, Determining the stability number of a graph, SIAM J. Comput. 6 (1977) 643.

    Google Scholar 

  10. V. Chvátal, Hard knapsack problems, Oper. Res. 28 (1980) 1402.

    Google Scholar 

  11. D.G. Corneil and B. Graham, An algorithm for determining the chromatic number of a graph, SIAM J. Comput. 2 (1973) 311.

    Google Scholar 

  12. F.D.J. Dunstan, Sequential colourings of graphs, Proc. 5th British Comb. Conf., ed. C. St. J.A. Nash-Williams and J. Sheehan (Utilitas Mathematica, Winnipeg, 1976) p. 151.

    Google Scholar 

  13. K. Dürre, Approximate algorithms for coloring large graphs, Datenstrukt, Graphen, Algorithm 3 Fachtag Graphentheor. Konz. Inf., Linz 1977, (1978) p. 191.

  14. P. Erdős, Graph theory and probability, Can. J. Math. 11 (1959) 34.

    Google Scholar 

  15. P. Erdős, On circuits and subgraphs of random graphs, Mathematika 9 (1962) 170.

    Google Scholar 

  16. P. Erdős and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960) 17.

    Google Scholar 

  17. P. Erdős and J. Spencer, Probabilistic Methods in Combinatorics (Academic Press, New York and London, 1974).

    Google Scholar 

  18. M.R. Garey and D.S. Johnson, The complexity of near optimal graph coloring, J.A.C.M. 23 (1976) 43.

    Google Scholar 

  19. M.R. Garey and D.S. Johnson, Computers and Intractability (Freeman, San Francisco, 1979).

    Google Scholar 

  20. G.R. Grimmett, Random graph theorems, Trans. 7th Prague Conf. on Information Th. and related topics A (1977) p. 203.

  21. G.R. Grimmett, Random graphs, in: Selected Topics in Graph Theory 2, ed. L. Beineke and R. Wilson (Academic Press, London, 1983) p. 201.

    Google Scholar 

  22. G.R. Grimmett and C.J.H. McDiarmid, On colouring random graphs, Math. Proc. Camb. Phil. Soc. 77 (1975) 313.

    Google Scholar 

  23. D. Hochbaum, Easy solutions for theK-center problem or the dominating set problem on random graphs (1982) preprint.

  24. D.S. Johnson, Worst case behaviour of graph colouring algorithms, Proc. 5th S.E. Conf. on Combinatorics, Graph Theory and Computing (Utilitas Mathematica, Winnipeg, 1974) p. 513.

    Google Scholar 

  25. A. Johri and D.W. Matula, Probabilistic bounds and heuristic algorithms for coloring large random graphs, Technical Report 82-CSE-6 (Southern Methodist University, Dallas, Texas, 1982).

    Google Scholar 

  26. M. Karonski, A review of random graphs, J. Graph. Th. 6 (1982) 349.

    Google Scholar 

  27. R.M. Karp, The probabilistic analysis of some combinatorial search algorithms, in: Algorithms and Complexity, ed. J.F. Traub (Academic Press, New York, 1976) p. 1.

    Google Scholar 

  28. T. Kawaguchi, H. Nakano and Y. Nakanishi, Probabilistic analysis of a heuristic graph coloring algorithm (1982) preprint.

  29. D.G. Kelly and J.G. Oxley, Asymptotic properties of random subsets of projective spaces, Math. Proc. Camb. Phil. Soc. 91 (1982) 119.

    Google Scholar 

  30. A.D. Korsunov, The chromatic number ofn-vertex graphs, Metody Diskret. Analiz. No. 35 (1980) 14–44, 104 (in Russian).

    Google Scholar 

  31. L. Kućera, Expected behaviour of graph coloring algorithms, Lecture Notes in Computer Science 56 (1977) p. 447.

    Google Scholar 

  32. E.L. Lawler, A note on the complexity of the chromatic number problem, Inf. Proc. Lett. 5 (1976) 66.

    Google Scholar 

  33. F.T. Leighton, A graph coloring algorithm for large scheduling problems, J. Res. Nat. Bur. Stand. 84 (1979) 489.

    Google Scholar 

  34. D.W. Matula, On the complete subgraphs of a random graph, Comb. Math. and its Appls. (Chapel Hill, N.C., 1970) p. 356.

  35. D.W. Matula, The employee party problem, Not. A.M.S. 19 (1972) A-382.

    Google Scholar 

  36. D.W. Matula, private communication (1982).

  37. D.W. Matula, G. Marble and J.D. Isaacson, Graph colouring algorithms, in: Graph Theory and Computing, ed. R.C. Read (Academic Press, New York, 1972) p. 109.

    Google Scholar 

  38. C.J.H. McDiarmid, Determining the chromatic number of a graph, Technical Report STAN-CS-76-576 (Stanford University, 1976).

  39. C.J.H. McDiarmid, Determining the chromatic number of a graph, SIAM J. Comput. 8 (1979) 1.

    Google Scholar 

  40. C.J.H. McDiarmid, Colouring random graphs badly, in: Graph Theory and Combinatorics, ed. R.J. Wilson (Pitman Research Notes in Mathematics 34, Pitman, London, 1979) p. 76.

    Google Scholar 

  41. C.J.H. McDiarmid, Achromatic numbers of random graphs, Math. Proc. Camb. Phil. Soc. 92 (1982) 21.

    Google Scholar 

  42. C.J.H. McDiarmid, On the chromatic forcing number of a random graph, Discr. Appl. Math. 5 (1983) 123.

    Google Scholar 

  43. D.M. Miller, An algorithm for determining the chromatic number of a graph, Proc. 5th Manitoba Conf. on Numerical Math., ed. Hartnell and Williams (1975) p. 533.

  44. J. Mitchem, On various algorithms for estimating the chromatic number of a graph, Comput. J. 19 (1976) 182.

    Google Scholar 

  45. A. Punter, private communications.

  46. G. Schmidt and T. Ströhleim, Timertable construction — an annotated bibliography, Comput. J. 23 (1980) 307.

    Google Scholar 

  47. J. Schmidt-Pruzan, Probabilistic analysis of strong hypergraph coloring algorithms and the strong chromatic number (1983) preprint.

  48. J. Schmidt-Pruzan, E. Shamir and E. Upfal, Random hypergraph coloring algorithms and the weak chromatic number (1983) preprint.

  49. E. Shamir and E. Upfal, Sequential and distributed graph coloring algorithms with performance analysis in random graph spaces (1982) preprint.

  50. G. Tinhofer, Zufallsgraphen (Applied Computer Science 17, Hanser, Munich and Vienna, 1980).

    Google Scholar 

  51. I. Tomescu, On the chromatic number of almost all graphs, Bull. Math. de la Soc. Sci. Math. de la R.S. de Roumanie Tome 25(73), nr. 3 (1981) 321.

    Google Scholar 

  52. W.F. de la Vega, On the chromatic number of sparse random graphs (1983) preprint.

  53. W.F. de la Vega, Crowded graphs can be colored within 1+ε in polynomial time (1983) preprint.

  54. C.C. Wang, An algorithm for the chromatic number of a graph, J.A.C.M. 21 (1974) 385.

    Google Scholar 

  55. A. Wigderson, A new approximate graph coloring algorithm (1983) preprint.

  56. H.S. Wilf, Backtrack: an O(1) expected time algorithm for the graph coloring problem (1983) preprint.

  57. B. Pittel, On the probable behaviour of some algorithms for finding the stability number of a graph, Math. Proc. Camb. Phil. Soc. 92 (1982) 511.

    Google Scholar 

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  1. Institute of Economics and Statistics, University of Oxford, St. Cross Building Manor Road, OX1 3UL, Oxford, England

    Colin McDiarmid

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  1. Colin McDiarmid
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McDiarmid, C. Colouring random graphs. Ann Oper Res 1, 183–200 (1984). https://doi.org/10.1007/BF01874388

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