On the connectivity threshold for colorings of random graphs and hypergraphs

TL;DR

This paper investigates the connectivity of the space of proper colorings in random hypergraphs, establishing conditions under which the coloring graph is connected based on hypergraph density and number of colors.

Contribution

It provides a threshold for the connectivity of the coloring graph in random hypergraphs, linking hypergraph parameters to the colorings' connectivity.

Findings

Connectivity of coloring graph w.h.p. for large hypergraph density

Threshold for number of colors based on hypergraph degree and uniformity

Conditions under which the coloring space is connected

Abstract

Let $\Omega_q=\Omega_q(H)$ denote the set of proper $[q]$-colorings of the hypergraph $H$. Let $\Gamma_q$ be the graph with vertex set $\Omega_q$ and an edge ${\sigma,\tau\}$ where $\sigma,\tau$ are colorings iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|\{v\in V(H):\sigma(v)\neq\tau(v)\}|$. We show that if $H=H_{n,m;k},\,k\geq 2$, the random $k$-uniform hypergraph with $V=[n]$ and $m=dn/k$ then w.h.p. $\Gamma_q$ is connected if $d$ is sufficiently large and $q\gtrsim (d/\log d)^{1/(k-1)}$.