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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Combinatorial probability extremal combinatorics graph coloring limit theorems


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.

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