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February 2017 Universal limit theorems in graph coloring problems with connections to extremal combinatorics
Bhaswar B. Bhattacharya, Persi Diaconis, Sumit Mukherjee
Ann. Appl. Probab. 27(1): 337-394 (February 2017). DOI: 10.1214/16-AAP1205

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.

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Bhaswar B. Bhattacharya. Persi Diaconis. Sumit Mukherjee. "Universal limit theorems in graph coloring problems with connections to extremal combinatorics." Ann. Appl. Probab. 27 (1) 337 - 394, February 2017. https://doi.org/10.1214/16-AAP1205

Information

Received: 1 November 2015; Revised: 1 April 2016; Published: February 2017
First available in Project Euclid: 6 March 2017

zbMATH: 1360.05051
MathSciNet: MR3619790
Digital Object Identifier: 10.1214/16-AAP1205

Subjects:
Primary: 05C15 , 60C05 , 60F05
Secondary: 05D99

Keywords: combinatorial probability , extremal combinatorics , Graph coloring , limit theorems

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 1 • February 2017
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