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On the concentration of the chromatic number of random graphs
Published 3 Jan 2022 in math.CO, cs.DM, and math.PR | (2201.00906v2)
Abstract: Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.
- D. Achlioptas and A. Naor. The two possible values of the chromatic number of a random graph. Ann. of Math. 162 (2005), 1335–1351.
- N. Alon and M. Krivelevich. The concentration of the chromatic number of random graphs. Combinatorica 17 (1997), 303–313.
- N. Alon and J. Spencer. The Probabilistic Method, 2nd ed. Wiley-Interscience, New York (2000).
- B. Bollobás. The chromatic number of random graphs. Combinatorica 8 (1988), 49–55.
- B. Bollobás. How sharp is the concentration of the chromatic number? Combin. Probab. Comput. 13 (2004), 115–117.
- G. Grimmett and C.McDiarmid. On colouring random graphs. Math. Proc. Cambridge Philos. Soc. 77 (1975), 313–324.
- Prague dimension of random graphs. Combinatorica 43 (2023), 853–884.
- A. Heckel. Non-concentration of the chromatic number of a random graph. J. Amer. Math. Soc. 34 (2021), 245–260.
- A. Heckel and O. Riordan. J. London Math. Soc. 108 (2023), 1769–1815.
- M. Isaev and M. Kang, On the chromatic number in the stochastic block model. Electron. J. Combin. 30 (2023), Paper 2.56, 50pp.
- S. Janson. Poisson approximation for large deviations. Random Struct. Alg. 1 (1990), 221–229.
- Random Graphs. Wiley-Interscience, New York (2000).
- A. Johansson, J. Kahn and V. Vu. Factors in Random Graphs. Random Struct. Alg. 33 (2008), 1–28.
- R. Kang and C. McDiarmid. Colouring random graphs. In Topics in chromatic graph theory, pp. 199–229, Cambridge Univ. Press, Cambridge (2015).
- M. Krivelevich. On the minimal number of edges in color-critical graphs. Combinatorica 17 (1997), 401-426.
- T. Łuczak. The chromatic number of random graphs. Combinatorica 11 (1991), 45–54.
- T. Łuczak. A note on the sharp concentration of the chromatic number of random graphs. Combinatorica 11 (1991), 295–297.
- The jump of the clique chromatic number of random graphs. Random Struct. Alg. 62 (2023), 1016–1034.
- C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, pp. 148–188. Cambridge Univ. Press, Cambridge (1989).
- Clique coloring of binomial random graphs. Random Struct. Alg. 54 (2019), 589–614.
- C. Mcdiarmid and B. Reed. Concentration for Self-bounding Functions and an inequality of Talagrand. Random Struct. Alg. 29 (2006), 549–557.
- O. Riordan and L. Warnke. The Janson inequalities for general up-sets. Random Struct. Alg. 46 (2015), 391–395.
- A. Ruciński. When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields 78 (1988), 1–10.
- A. Ruciński. Matching and covering the vertices of a random graph by copies of a given graph. Discrete Math. 105 (1992), 185–197.
- A. Scott. On the concentration of the chromatic number of random graphs. Explanatory note (2008). arXiv:0806.0178
- E. Shamir and J. Spencer. Sharp concentration of the chromatic number on random graphs Gn,psubscript𝐺𝑛𝑝G_{n,p}italic_G start_POSTSUBSCRIPT italic_n , italic_p end_POSTSUBSCRIPT. Combinatorica 7 (1987), 121–129.
- L. Warnke. On the method of typical bounded differences. Combin. Probab. Comput. 25 (2016), 269–299.
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