{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HBCGMLERB7LTDJW6HD5IT7FM3B","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8a9492c3725bb64d79aa9bb5d4e604e76544968dee9bfd25d36537323a87e82d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-29T00:02:40Z","title_canon_sha256":"c6d35cc83b86c982ba288f6158746d1bae074998fa8657c5a1dff07a2022ba6c"},"schema_version":"1.0","source":{"id":"1205.6240","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.6240","created_at":"2026-05-18T03:20:09Z"},{"alias_kind":"arxiv_version","alias_value":"1205.6240v3","created_at":"2026-05-18T03:20:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.6240","created_at":"2026-05-18T03:20:09Z"},{"alias_kind":"pith_short_12","alias_value":"HBCGMLERB7LT","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HBCGMLERB7LTDJW6","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HBCGMLER","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:51725f53e8e118cd93d751c85c3c30276f979729c8ca2da5ed3e258553328ee8","target":"graph","created_at":"2026-05-18T03:20:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a finite graph with minimum degree $r$. Form a random subgraph $G_p$ of $G$ by taking each edge of $G$ into $G_p$ independently and with probability $p$. We prove that for any constant $\\epsilon>0$, if $p=\\frac{1+\\epsilon}{r}$, then $G_p$ is non-planar with probability approaching 1 as $r$ grows. This generalizes classical results on planarity of binomial random graphs.","authors_text":"Alan Frieze, Michael Krivelevich","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-29T00:02:40Z","title":"On the non-planarity of a random subgraph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.6240","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3048319d8071fac4be853558867e90687d73ff69b4b3fb65351dbedf7e5242a8","target":"record","created_at":"2026-05-18T03:20:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8a9492c3725bb64d79aa9bb5d4e604e76544968dee9bfd25d36537323a87e82d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-29T00:02:40Z","title_canon_sha256":"c6d35cc83b86c982ba288f6158746d1bae074998fa8657c5a1dff07a2022ba6c"},"schema_version":"1.0","source":{"id":"1205.6240","kind":"arxiv","version":3}},"canonical_sha256":"3844662c910fd731a6de38fa89fcacd86a7b1f1e6c4858a4156e8a4fdc224d52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3844662c910fd731a6de38fa89fcacd86a7b1f1e6c4858a4156e8a4fdc224d52","first_computed_at":"2026-05-18T03:20:09.869847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:09.869847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xQV6bnhsZRuhnyT2gW4tuLapBmTA3QVfFpS2TXrIxMggF6IrA6FnooJ5a435rzhNipkICkgBWB6yvJuF+Jl2Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:09.870310Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.6240","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3048319d8071fac4be853558867e90687d73ff69b4b3fb65351dbedf7e5242a8","sha256:51725f53e8e118cd93d751c85c3c30276f979729c8ca2da5ed3e258553328ee8"],"state_sha256":"b1d5f828a0eedb73f428e160007862801e294cd25996dcef9d67013b6774ec2c"}