{"paper":{"title":"On the Rigidity of Sparse Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Jonathan Mosheiff, Nati Linial","submitted_at":"2015-05-05T21:20:06Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01189","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}