# Limit Theorems for Monochromatic Stars

## TL;DR

This paper characterizes the limiting distribution of monochromatic star counts in random graph colorings, revealing a sum of independent polynomial Poisson components, with implications for understanding graph coloring patterns.

## Contribution

It provides a complete characterization of the limiting distribution of monochromatic star counts in random graph colorings, including a representation as sums of polynomial Poisson variables.

## Key findings

- Limiting distribution is a sum of independent polynomial Poisson variables.
- Characterization applies to any growing sequence of graphs with bounded expected counts.
- Connections to classical problems like the birthday problem are discussed.

## Abstract

Let $T(K_{1, r}, G_n)$ be the number of monochromatic copies of the $r$-star $K_{1, r}$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we provide a complete characterization of the limiting distribution of $T(K_{1, r}, G_n)$, in the regime where $\mathbb E(T(K_{1, r}, G_n))$ is bounded, for any growing sequence of graphs $G_n$. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most $r$. Conversely, any limiting distribution of $T(K_{1, r}, G_n)$ has a representation of this form. Examples and connections to the birthday problem are discussed.