The Annals of Applied Probability

Universal limit theorems in graph coloring problems with connections to extremal combinatorics

Bhaswar B. Bhattacharya, Persi Diaconis, and Sumit Mukherjee

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Abstract

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. These can be seen as generalizations of the birthday problem (what is the chance that there are two friends with the same birthday?). It is shown that if the number of colors grows to infinity, the asymptotic distribution is either a Poisson mixture or a Normal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors. This result holds for any graph sequence, deterministic or random. On the other hand, when the number of colors is fixed, a necessary and sufficient condition for asymptotic normality is determined. Finally, using some results from the emerging theory of dense graph limits, the asymptotic (nonnormal) distribution is characterized for any converging sequence of dense graphs. The proofs are based on moment calculations which relate to the results of Erdős and Alon on extremal subgraph counts. As a consequence, a simpler proof of a result of Alon, estimating the number of isomorphic copies of a cycle of given length in graphs with a fixed number of edges, is presented.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 337-394.

Dates
Received: November 2015
Revised: April 2016
First available in Project Euclid: 6 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1488790830

Digital Object Identifier
doi:10.1214/16-AAP1205

Mathematical Reviews number (MathSciNet)
MR3619790

Zentralblatt MATH identifier
1360.05051

Subjects
Primary: 05C15: Coloring of graphs and hypergraphs 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Secondary: 05D99: None of the above, but in this section

Keywords
Combinatorial probability extremal combinatorics graph coloring limit theorems

Citation

Bhattacharya, Bhaswar B.; Diaconis, Persi; Mukherjee, Sumit. Universal limit theorems in graph coloring problems with connections to extremal combinatorics. Ann. Appl. Probab. 27 (2017), no. 1, 337--394. doi:10.1214/16-AAP1205. https://projecteuclid.org/euclid.aoap/1488790830


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References

  • [1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. Springer, New York.
  • [2] Alon, N. (1981). On the number of subgraphs of prescribed type of graphs with a given number of edges. Israel J. Math. 38 116–130.
  • [3] Alon, N. (1986). On the number of certain subgraphs contained in graphs with a given number of edges. Israel J. Math. 53 97–120.
  • [4] Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5 403–434.
  • [5] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • [6] Beran, R. (1975). Tail probabilities of noncentral quadratic forms. Ann. Statist. 3 969–974.
  • [7] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
  • [8] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
  • [9] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
  • [10] Burda, M., Harding, M. and Hausman, J. (2012). A Poisson mixture model of discrete choice. J. Econometrics 166 184–203.
  • [11] Cerquetti, A. and Fortini, S. (2006). A Poisson approximation for coloured graphs under exchangeability. Sankhyā 68 183–197.
  • [12] Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
  • [13] Chatterjee, S. and Diaconis, P. (2013). Estimating and understanding exponential random graph models. Ann. Statist. 41 2428–2461.
  • [14] Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64–106.
  • [15] Chung, F. R. K., Graham, R. L. and Wilson, R. M. (1989). Quasi-random graphs. Combinatorica 9 345–362.
  • [16] Church, K. and Gale, W. A. (1995). Poisson mixtures. Nat. Lang. Eng. 1 163–190.
  • [17] Conlon, D., Fox, J. and Zhao, Y. (2014). Extremal results in sparse pseudorandom graphs. Adv. Math. 256 206–290.
  • [18] DasGupta, A. (2005). The matching, birthday and the strong birthday problem: A contemporary review. J. Statist. Plann. Inference 130 377–389.
  • [19] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277.
  • [20] Diaconis, P. and Holmes, S. (2002). A Bayesian peek into Feller volume I. Sankhyā Ser. A 64 820–841. Special issue in memory of D. Basu.
  • [21] Diaconis, P. and Mosteller, F. (1989). Methods for studying coincidences. J. Amer. Statist. Assoc. 84 853–861.
  • [22] Dong, F. M., Koh, K. M. and Teo, K. L. (2005). Chromatic Polynomials and Chromaticity of Graphs. World Scientific Co. Pte. Ltd., Hackensack, NJ.
  • [23] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
  • [24] Erdős, P. (1962). On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 7 459–464.
  • [25] Fadnavis, S. (2011). A generalization of the birthday problem and the chromatic polynomial. Preprint. Available at arXiv:1105.0698v2.
  • [26] Fang, X. (2015). A universal error bound in the CLT for counting monochromatic edges in uniformly colored graphs. Electron. Commun. Probab. 20 21.
  • [27] Friedgut, E. and Kahn, J. (1998). On the number of copies of one hypergraph in another. Israel J. Math. 105 251–256.
  • [28] Gelman, A. (2013). 600: The average American knows how many people? New York Times February 19, page D7.
  • [29] Götze, F. and Tikhomirov, A. (2002). Asymptotic distribution of quadratic forms and applications. J. Theoret. Probab. 15 423–475.
  • [30] Götze, F. and Tikhomirov, A. N. (1999). Asymptotic distribution of quadratic forms. Ann. Probab. 27 1072–1098.
  • [31] Greenwood, M. and Yule, G. V. (1920). An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attack of disease or of repeated accidents. J. Roy. Statist. Soc. Ser. A 83 255–279.
  • [32] Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal. 14 1–16.
  • [33] Janson, S., Oleszkiewicz, K. and Ruciński, A. (2004). Upper tails for subgraph counts in random graphs. Israel J. Math. 142 61–92.
  • [34] Jensen, T. R. and Toft, B. (1995). Graph Coloring Problems. Wiley, New York.
  • [35] Jensen, T. R. and Toft, B. (2013). Unsolved graph coloring problems. Manuscript in preparation.
  • [36] Killworth, P. and Bernard, H. (1979). The reversal small-world experiment. Social Networks 1 159–192.
  • [37] Le Cam, L. and Traxler, R. (1978). On the asymptotic behavior of mixtures of Poisson distributions. Z. Wahrsch. Verw. Gebiete 44 1–45.
  • [38] Lovász, L. (2012). Large Networks and Graph Limits. American Mathematical Society Colloquium Publications 60. Amer. Math. Soc., Providence, RI.
  • [39] Nemhauser, G. L. and Trotter, L. E. Jr. (1974). Properties of vertex packing and independence system polyhedra. Math. Program. 6 48–61.
  • [40] Nourdin, I., Peccati, G., Poly, G. and Simone, R. (2016). Classical and free fourth moment theorems: Universality and thresholds. J. Theoret. Probab. 29 653–680.
  • [41] Nourdin, I., Peccati, G. and Reinert, G. (2010). Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15 1703–1742.
  • [42] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947–1985.
  • [43] Rotar, V. I. (1973). Some limit theorems for polynomials of second degree. Theory Probab. Appl. 18 499–507.
  • [44] Schrijver, A. (2003). Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics 24. Springer, Berlin.
  • [45] Sethuraman, J. (1961). Some limit theorems for joint distributions. Sankhyā Ser. A 23 379–386.
  • [46] Stanley, R. P. (1995). A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111 166–194.
  • [47] Sweeting, T. J. (1989). On conditional weak convergence. J. Theoret. Probab. 2 461–474.