Isomorphisms between random graphs

Abstract

Consider two independent Erdős–Rényi G(N,1/2) graphs. We show that with probability tending to 1 as N→∞, the largest induced isomorphic subgraph has size either ⌊xN−εN⌋ or ⌊xN+εN⌋, where xN=4log2⁡N−2log2⁡log2⁡N−2log2⁡(4/e)+1 and εN=(4log2⁡N)−1/2. Using similar techniques, we also show that if Γ1 and Γ2 are independent G(n,1/2) and G(N,1/2) random graphs, then Γ2 contains an isomorphic copy of Γ1 as an induced subgraph with high probability if n≤⌊yN−εN⌋ and does not contain an isomorphic copy of Γ1 as an induced subgraph with high probability if n>⌊yN+εN⌋, where yN=2log2⁡N+1 and εN is as above.