On the Rigidity of Sparse Random Graphs

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and for $p<\frac{(1-\epsilon)\log n}{n}$ with positive probability $\text{aut}(G)$ is nontrivial. We show that in the sparser case $\omega(\frac 1 n)\leq p\leq \frac{\log n}{n}+\omega(\frac 1n)$, it holds whp that $G$'s $2$-core is rigid. We conclude that for all $p$, a graph in $G(n,p)$ is reconstrutible whp. In addition this yields for $\omega(\frac 1n)\leq p\leq \frac 12$ a canonical labeling algorithm that almost surely runs in polynomial time with $o(1)$ error rate. This extends the range for which such an algorithm is currently known.