Isomorphisms between random graphs

TL;DR

This paper determines the asymptotic size of the largest induced isomorphic subgraph between two independent Erdős-Rényi graphs and characterizes when one random graph contains an induced copy of another.

Contribution

It provides precise probabilistic thresholds for the size of the largest induced isomorphic subgraph and for the existence of an induced isomorphic copy of one graph within another, using novel techniques.

Findings

Largest induced isomorphic subgraph size concentrates around a specific logarithmic function of N.

Thresholds for induced isomorphic copies depend on logarithmic functions of N.

High probability results for the existence or non-existence of induced isomorphic subgraphs.

Abstract

Consider two independent Erd\H{o}s-R\'enyi $G(N,1/2)$ graphs. We show that with probability tending to $1$ as $N\to\infty$, the largest induced isomorphic subgraph has size either $\lfloor x_N-\varepsilon_N\rfloor$ or $\lfloor x_N+\varepsilon_N \rfloor$, where $x_N=4\log_2 N -2 \log_2 \log_2 N - 2\log_2(4/e)+1$ and $\varepsilon_N = (4\log_2 N)^{-1/2}$. Using similar techniques, we also show that if $\Gamma_1$ and $\Gamma_2$ are independent $G(n,1/2)$ and $G(N,1/2)$ random graphs, then $\Gamma_2$ contains an isomorphic copy of $\Gamma_1$ as an induced subgraph with high probability if $n\le \lfloor y_N - \varepsilon_N \rfloor$ and does not contain an isomorphic copy of $\Gamma_1$ as an induced subgraph with high probability if $n>\lfloor y_N+\varepsilon_N \rfloor$, where $y_N=2\log_2 N+1$ and $\varepsilon_N$ is as above.