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10 - Colouring random graphs

Published online by Cambridge University Press:  05 May 2015

By
Ross J. Kang  and
Colin McDiarmid
Edited by
Lowell W. Beineke  and
Robin J. Wilson
Ross J. Kang
Affiliation:
University of Oxford
Colin McDiarmid
Affiliation:
University of Oxford
Lowell W. Beineke
Affiliation:
Purdue University, Indiana
Robin J. Wilson
Affiliation:
The Open University, Milton Keynes
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Summary

Typically how many colours are required to colour a graph? In other words, given a graph chosen randomly, what can we expect its chromatic number to be? We survey the classic interpretation of this question, with the binomial or Erdős–Rényi random graph and the usual chromatic number. We also treat a few variations, not only of the random graph model, but also of the chromatic parameter.

Introduction

How many colours are typically necessary to colour a graph?

We survey a number of perspectives on this natural question, which is central to random graph theory and to probabilistic and extremal combinatorics. It has stimulated a vibrant area of research, with a rich history extending back through more than half a century.

Erdős and Rényi [36] asked a form of this question in a celebrated early paper on random graphs in 1960. Let Gn,m be a graph chosen uniformly at random from the set of graphs with vertex-set [n] = {1, 2, …, n} and m edges. In this probabilistic model, we cannot rule out the possibility that Gn,m is, for example, the disjoint union of one large clique and some isolated vertices, or perhaps one Turán graph (a balanced complete multipartite graph) and some isolated vertices. The resulting range is large: in the former situation the chromatic number could be about, while in the latter it could be 2 if m ≤ n2/4. These outcomes are unlikely, however, and we are interested in the most probable ones. To state the question properly, we say that an event An (which here always describes a property of a random graph on the vertex-set [n]) holds asymptotically almost surely (a.a.s.) if the probability that An holds satisfies ℙ(An) → 1 as n → ∞. Erdős and Rényi asked the following question.

Type
Chapter
Information
Topics in Chromatic Graph Theory , pp. 199 - 229
Publisher: Cambridge University Press
Print publication year: 2015

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References

1. D., Achlioptas and E., Friedgut, A sharp threshold for k-colorability, Random Structures Algorithms 14 (1999), 63–70.Google Scholar
2. D., Achlioptas and C., Moore, The chromatic number of random regular graphs, Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX) and 8th International Workshop on Randomization and Computation (RANDOM), Lecture Notes in Computer Science, Springer (2004), 219–228.Google Scholar
3. D., Achlioptas and A., Naor, The two possible values of the chromatic number of a random graph, Ann. of Math. (2) 162 (2005), 1335–1351.Google Scholar
4. N., Alon, Choice numbers of graphs: a probabilistic approach, Combin. Probab. Comput. 1 (1992), 107–114.Google Scholar
5. N., Alon, Restricted colorings of graphs, Surveys in Combinatorics, 1993 (Keele) (ed. K., Walker), London Math. Soc. Lecture Notes 187, Cambridge Univ. Press (1993), 1–33.Google Scholar
6. N., Alon, The chromatic number of random Cayley graphs, Europ. J. Combin. 34 (2013), 232–243.Google Scholar
7. N., Alon and M., Krivelevich, The concentration of the chromatic number of random graphs, Combinatorica 17 (1997), 303–313.Google Scholar
8. N., Alon, M., Krivelevich and B., Sudakov, List coloring of random and pseudo-random graphs, Combinatorica 19 (1999), 453–472.Google Scholar
9. N., Alon and J., Spencer, The Probabilistic Method (3rd edn.), with an appendix on the life and work of Paul Erdős, Wiley–Interscience Series in Discrete Mathematics and Optimization, 2008.Google Scholar
10. S., Ben-Shimon and M., Krivelevich, Random regular graphs of non-constant degree: concentration of the chromatic number, Discrete Math. 309 (2009), 4149–4161.Google Scholar
11. E. A., Bender and E. R., Canfield, The asymptotic number of labeled graphs with given degree sequences, J. Combin. Theory (A) 24 (1978), 296–307.Google Scholar
12. E. A., Bender, Z., Gao and N. C., Wormald, The number of labeled 2-connected planar graphs, Electron. J. Combin. 9, Research Paper 43, 13 pp. (electronic), 2002.Google Scholar
13. B., Bollobas, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Europ. J. Combin. 1 (1980), 311–316.Google Scholar
14. B., Bollobas, The chromatic number of random graphs, Combinatorica 8 (1988), 49–55.Google Scholar
15. B., Bollobás, Random Graphs (2nd edn.), Cambridge Studies in Advanced Math. 73, 2001.Google Scholar
16. B., Bollobás, How sharp is the concentration of the chromatic number?, Combin. Probab. Comput. 13 (1) (2004), 115–117.Google Scholar
17. B., Bollobás and P., Erdős, Cliques in random graphs, Math. Proc. Cambridge Philos. Soc. 80 (1976), 419–427.Google Scholar
18. B., Bollobas and A., Thomason, Threshold functions, Combinatorica 7 (1987), 35–38.Google Scholar
19. B., Bollobás and A., Thomason, Generalized chromatic numbers of random graphs. Random Structures Algorithms 6 (1995), 353–356.Google Scholar
20. B., Bollobás and A., Thomason, The structure of hereditary properties and colourings of random graphs, Combinatorica 20 (2000), 173–202.Google Scholar
21. A., Braunstein, R., Mulet, A., Pagnani, M., Weigt and R., Zecchina, Polynomial iterative algorithms for coloring and analyzing random graphs, Phys. Rev. E (3) 68 (2003), 036702, 15.Google Scholar
22. G., Chapuy, E, Fusy, O, Gimenez, B., Mohar and M, Noy., Asymptotic enumeration and limit laws for graphs of fixed genus, J. Combin. Theory (A) 118 (2011), 748–777.Google Scholar
23. V., Chvátal, Almost all graphs with 1.44n edges are 3-colorable, Random Structures Algorithms 2 (1991), 11–28.Google Scholar
24. A., Coja-Oghlan, Upper-bounding the k-colorability threshold by counting covers, Elec-tron. J. Combin. 20 (2013), paper 32, 28.Google Scholar
25. A., Coja-Oghlan, C., Efthymiou and S., Hetterich, On the chromatic number of random regular graphs, ArXiv e-prints, Aug. 2013.
26. A., Coja-Oghlan, K., Panagiotou and A., Steger, On the chromatic number of random graphs, J. Combin. Theory (B) 98 (2008), 980–993.Google Scholar
27. A., Coja-Oghlan and D., Vilenchik, Chasing the k-colorability threshold, ArXiv e-prints, Apr. 2013.
28. C., Cooper, A., Frieze, B., Reed and O., Riordan, Random regular graphs of non-constant degree: independence and chromatic number, Combin. Probab. Comput. 11 (2002), 323–341.Google Scholar
29. M., DeVos, K., Kawarabayashi and B., Mohar, Locally planar graphs are 5-choosable, J. Combin. Theory (B) 98 (2008), 1215–1232.Google Scholar
30. J., Díaz, A. C., Kaporis, G. D., Kemkes, L. M., Kirousis, X. Pérez-Giménez and N. C. Wormald, On the chromatic number of a random 5-regular graph, J. Graph Theory 61 (2009), 157–191.Google Scholar
31. C., Dowden, Uniform Random Planar Graphs with Degree Constraints, Ph.D. thesis, University of Oxford, 2008, http://ora.ox.ac.uk/objects/uuid%3Af8a9afe3-30ad-4672-9a6c-4fb9ac9af041.
32. C., Dowden, The evolution of uniform random planar graphs, Electron. J. Combin. 17 (2010), R7, 20 pp.Google Scholar
33. M., Dyer, A., Frieze and C., Greenhill, On the chromatic number of a random hypergraph, ArXiv e-prints, Aug. 2012.
34. P., Erdős, Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.Google Scholar
35. P., Erdős and S., Fajtlowicz, On the conjecture of Hajs, Combinatorica 1 (1981), 141–143.Google Scholar
36. P., Erdős and A., Renyi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.Google Scholar
37. P., Erdős and R. J., Wilson, On the chromatic index of almost all graphs, J. Combin. Theory (B) 23 (1977), 255–257.Google Scholar
38. N., Fountoulakis, R. J., Kang and C., McDiarmid, The t-stability number of a random graph, Electron. J. Combin. 17 (2010), R59, 29 pp.Google Scholar
39. N., Fountoulakis, R. J., Kang and C., McDiarmid, Largest sparse subgraphs of random graphs, Europ. J. Combin. 35 (2014), 232–244.Google Scholar
40. E., Friedgut, Sharp thresholds of graph properties, and the k-SAT problem, with an appendix by Jean Bourgain, J. Amer. Math. Soc. 12 (1999), 1017–1054.Google Scholar
41. A., Frieze, On the independence number of random graphs, Discrete Math. 81 (1990), 171–175.Google Scholar
42. A., Frieze, B., Jackson, C., McDiarmid and B., Reed, Edge-colouring random graphs, J. Combin. Theory (B) 45 (1988), 135–149.Google Scholar
43. A., Frieze and T., Luczak, On the independence and chromatic numbers of random regular graphs, J. Combin. Theory (B) 54 (1992), 123–132.Google Scholar
44. A., Frieze and E., Vigoda, A survey on the use of Markov chains to randomly sample colourings, Combinatorics, Complexity, and Chance (eds. G., Grimmett and C., McDiarmid), Oxford Lecture Ser. Math. Appl. 34, Oxford Univ. Press (2007), 53–71.Google Scholar
45. O., Giménez and M., Noy, Asymptotic enumeration and limit laws of planar graphs, J. Amer. Math. Soc. 22 (2009), 309–329.Google Scholar
46. B., Green, On the chromatic number of random Cayley graphs, ArXiv e-prints, Aug. 2013.
47. B., Green and R., Morris, Counting sets with small sumset and applications, ArXiv e-prints, May 2013.
48. G., Grimmett and C., McDiarmid, On colouring random graphs, Math. Proc. Cambridge Philos. Soc. 77 (1975), 313–324.Google Scholar
49. S., Janson, T., Łuczak and A., Ruciński, Random Graphs, Wiley-Interscience Series in Discrete Math. and Optimization, 2000.Google Scholar
50. M., Kang and T., Łuczak, Two critical periods in the evolution of random planar graphs, Trans. Amer. Math. Soc. 364 (2012), 4239–4265.Google Scholar
51. R. J., Kang and C., McDiarmid, The t-improper chromatic number of random graphs, Combin. Probab. Comput. 19 (2010), 87–98.Google Scholar
52. R. M., Karp, The probabilistic analysis of some combinatorial search algorithms, Algorithms and Complexity (Carnegie-Mellon Univ., 1976) (ed. J. F., Traub), Academic Press (1976), 1–19.
53. G., Kemkes, X., Pérez-Giménez and N. C., Wormald, On the chromatic number of random d-regular graphs, Adv. Math. 223 (2010), 300–328.Google Scholar
54. J. H., Kim and V. H., Vu, Sandwiching random graphs: universality between random graph models, Adv. Math. 188 (2004), 444–469.Google Scholar
55. M., Krivelevich, The choice number of dense random graphs, Combin. Probab. Comput. 9 (2000), 19–26.Google Scholar
56. M., Krivelevich and B., Patkόs, Equitable coloring of random graphs, Random Structures Algorithms 35 (2009), 83–99.Google Scholar
57. M., Krivelevich, B., Sudakov, V. H., Vu and N. C., Wormald, Random regular graphs of high degree, Random Structures Algorithms 18 (2001), 346–363.Google Scholar
58. M., Krivelevich, B., Sudakov, V. H., Vu and N. C., Wormald, On the probability of independent sets in random graphs, Random Structures Algorithms 22 (2003), 1–14.Google Scholar
59. T., Łuczak, The chromatic number of random graphs, Combinatorica 11 (1991), 45–54.Google Scholar
60. T., Łuczak, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (1991), 295–297.Google Scholar
61. E., Marchant and A., Thomason, The structure of hereditary properties and 2-coloured multigraphs, Combinatorica 31 (2011), 85–93.Google Scholar
62. D. W., Matula, On the complete subgraphs of a random graph, Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. of North Carolina, 1970), Univ. of North Carolina, Chapel Hill (1970), 356–369.Google Scholar
63. D. W., Matula, The employee party problem, Notices Amer. Math. Soc. 19 (1972), A–382.Google Scholar
64. D. W., Matula, Expose-and-merge exploration and the chromatic number of a random graph, Combinatorica 7 (1987), 275–284.Google Scholar
65. D. W., Matula and L., Kučera, An expose-and-merge algorithm and the chromatic number of a random graph, Random Graphs '87 (Poznanń, 1987) (eds. M., Karońiski, J., Jaworski and A., Ruciński), Wiley (1990), 175–187.Google Scholar
66. C., McDiarmid, Colouring random graphs badly, Graph Theory and Combinatorics (Open Univ., Milton Keynes, 1978) (ed. R. J., Wilson), Research Notes in Math. 34, Pitman (1979), 76–86.Google Scholar
67. C., McDiarmid, On the chromatic number of random graphs, Random Structures Algorithms 1 (1990), 435–442.Google Scholar
68. C., McDiarmid, Random channel assignment in the plane, Random Structures Algorithms 22 (2003), 187–212.Google Scholar
69. C., McDiarmid, Random graphs on surfaces, J. Combin. Theory (B) 98 (2008), 778–797.Google Scholar
70. C., McDiarmid, Random graphs from a minor-closed class, Combin. Probab. Comput. 18 (2009), 583–599.Google Scholar
71. C., McDiarmid, Random graphs from a weighted minor-closed class, Electron. J. Combin. 20 (2013), R52, 39 pp.Google Scholar
72. C., McDiarmid and T., Müller, On the chromatic number of random geometric graphs, Combinatorica 31 (2011), 423–488.Google Scholar
73. C., McDiarmid and B., Reed, On total colourings of graphs, J. Combin. Theory (B) 57 (1993), 122–130.Google Scholar
74. C., McDiarmid, A., Steger and D., Welsh, Random planar graphs, J. Combin. Theory (B) 93 (2005), 187–205.Google Scholar
75. M., Molloy and B., Reed, Graph Colouring and the Probabilistic Method, Algorithms and Combinatorics 23, Springer-Verlag, 2002.Google Scholar
76. R., Mulet, A., Pagnani, M., WeigtandR., Zecchina, Coloring randomgraphs, Phys. Rev. Lett. 89 (2002), 268701, 4.Google Scholar
77. T., Müller, Two-point concentration in random geometric graphs, Combinatorica 28 (2008), 529–545.Google Scholar
78. K., Panagiotou and A., Steger, A note on the chromatic number of a dense random graph, Discrete Math. 309 (2009), 3420–3423.Google Scholar
79. M., Penrose, Random Geometric Graphs, Oxford Studies in Probability 5, Oxford University Press, 2003.Google Scholar
80. M. P., Rombach and A., Scott, Equitable colourings of random graphs, in preparation, 2014.
81. E. R., Scheinerman, Generalized chromatic numbers of random graphs, SIAM J. Discrete Math. 5 (1992), 74–80.Google Scholar
82. E., Shamir and J., Spencer, Sharp concentration of the chromatic number on random graphs Gn,p, Combinatorica 7 (1987), 121–129.Google Scholar
83. L., Shi and N. C., Wormald, Colouring random 4-regular graphs, Combin. Probab. Comput. 16 (2007), 309–344.Google Scholar
84. L., Shi and N. C., Wormald, Colouring random regular graphs, Combin. Probab. Comput. 16 (2007), 459–494.Google Scholar
85. C., Thomassen, Five-coloring maps on surfaces, J. Combin. Theory (B) 59 (1993), 89–105.Google Scholar
86. V. H., Vu, On some simple degree conditions that guarantee the upper bound on the chromatic (choice) number of random graphs, J. Graph Theory 31 (1999), 201–226.Google Scholar
87. N. C., Wormald, Differential equations for random processes and random graphs, Ann. Appl. Probab. 5 (1995), 1217–1235.Google Scholar
88. N. C., Wormald, Models of random regular graphs, Surveys in Combinatorics, 1999 (Canterbury) (eds. J. D., Lamb and D. A., Preece), London Math. Soc. Lecture Notes 267, Cambridge Univ. Press (1999), 239–298.Google Scholar

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