# Constraining the clustering transition for colorings of sparse random graphs

## TL;DR

This paper investigates the structure of proper q-colorings in sparse random graphs, showing that above certain density thresholds, almost all colorings form a single giant component in the associated graph.

## Contribution

It establishes a new threshold for the clustering transition in the space of graph colorings, linking graph density and number of colors to the emergence of a giant component.

## Key findings

- A giant component exists in the coloring graph for large enough average degree d.
- The threshold for the clustering transition is characterized by q ≥ (c*d)/log d with c > 3/2.
- Almost all colorings are contained within a single giant component above this threshold.

## Abstract

Let $\Omega_q$ denote the set of proper $q$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $\Omega_q$ and an edge $\{\sigma,\tau\}$ where $\sigma,\tau$ are mappings $[n]\to[q]$ iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|\{v\in [n]:\sigma(v)\neq\tau(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $\Omega_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.