Isomorphisms between dense random graphs

05/08/2023

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by   Erlang Surya, et al.

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We consider two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities p_1,p_2. We resolve several open problems of Chatterjee and Diaconis, and also confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble: (i) we prove a sharp threshold result for the appearance of G_n,p_1 as an induced subgraph of G_N,p_2, (ii) we show two-point concentration of the maximum common induced subgraph of G_N, p_1 and G_N,p_2, and (iii) we show that the number of induced copies of G_n,p_1 in G_N,p_2 has an unusual limiting distribution.