The Second Moment Phenomenon for Monochromatic Subgraphs

TL;DR

This paper establishes that the distribution of monochromatic subgraphs in random colorings converges to a Poisson distribution based solely on the convergence of mean and variance, with specific conditions for different graph types.

Contribution

It proves a second-moment phenomenon for the distribution of monochromatic subgraphs, showing convergence to Poisson distribution when mean and variance match, and characterizes when this occurs.

Findings

Convergence to Poisson distribution occurs when mean and variance converge to the same limit.

The second-moment phenomenon holds if and only if the subgraph is a star-graph.

Multiple phase transitions are identified in Erdős-Rényi graphs depending on graph balance.

Abstract

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with $365$ colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show that $T(H, G_n)$ converges to $\mathrm{Pois}(\lambda)$ whenever $\mathbb E T(H, G_n) \rightarrow \lambda$ and $\mathrm{Var} T(H, G_n) \rightarrow \lambda$, that is, the asymptotic Poisson distribution of $T(H, G_n)$ is determined just by the convergence of its mean and variance. Moreover, this condition is necessary if and only if $H$ is a star-graph. In fact, the second-moment phenomenon is a consequence of a more general theorem about the convergence of $T(H,G_n)$ to a finite linear combination of independent Poisson random variables. As an application, we derive the limiting distribution of $T(H, G_n)$, when $G_n\sim G(n, p)$ is the Erd\H os-R\'enyi random graph. Multiple phase-transitions emerge as $p$ varies from 0 to 1, depending on whether the graph $H$ is balanced or unbalanced.