On the Rigidity of Sparse Random Graphs

A graph with a trivial automorphism group is said to be rigid. Wright proved (Acta Math 126(1) (1971), 1–9) that for log n/n + ω(1/n) d p d 1/2 a random graph G ϵG(n,p) is rigid whp (with high probability). It is not hard to see that this lower bound is sharp and for with positive probability is nontrivial. We show that in the sparser case aut (G), it holds whp that G's 2-core is rigid. We conclude that for all p, a graph in G(n,p) is reconstructible whp. In addition this yields for ω(1/n) d p d 1/2 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 (T. Czajka and G. Pandurangan, J Discrete Algorithms 6(1) (2008), 85–92).