On the Rigidity of Sparse Random Graphs

Nati Linial, Jonathan Mosheiff

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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).

Original languageAmerican English
Pages (from-to)466-480
Number of pages15
JournalJournal of Graph Theory
Issue number2
StatePublished - Jun 2017

Bibliographical note

Publisher Copyright:
© 2016 Wiley Periodicals, Inc.


  • core
  • graph canonical labeling
  • reconstruction
  • rigidity
  • sparse random graph


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