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On the connectivity threshold for colorings of random graphs and hypergraphs (1803.05246v2)
Published 14 Mar 2018 in math.CO, cs.DM, and math.PR
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 $q\gtrsim (d/\log d){1/(k-1)}$.