Statistics Papers

Document Type

Journal Article

Date of this Version

2017

Publication Source

The Annals of Applied Probability

Volume

27

Issue

1

Start Page

337

Last Page

394

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.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aoap/1488790830#abstract

Keywords

Combinatorial probability, extremal combinatorics, graph coloring, limit theorems

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Date Posted: 27 November 2017

This document has been peer reviewed.