On the connectivity threshold for colorings of random graphs and hypergraphs

03/14/2018

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by   Michael Anastos, et al.

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Let Ω_q=Ω_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Γ_q be the graph with vertex set Ω_q and an edge σ,τ} where σ,τ are colorings iff h(σ,τ)=1. Here h(σ,τ) is the Hamming distance |{v∈ V(H):σ(v)≠τ(v)}|. We show that if H=H_n,m;k, k≥ 2, the random k-uniform hypergraph with V=[n] and m=dn/k then w.h.p. Γ_q is connected if d is sufficiently large and q≳ (d/ d)^1/(k-1).