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Suppose the minimum degree of $\\Gamma$ is $\\delta(\\Gamma) \\geq (1/2 + \\varepsilon)n$ for some constant $\\varepsilon > 0$. Then with high probability, $\\Gamma_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two.\n  Given $0\\leq p\\leq 1$ we let $\\Gamma_p$ be the Erd\\H{o}s-R\\'{e}nyi subgraph of $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1811.03501","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-08T15:43:19Z","cross_cats_sorted":[],"title_canon_sha256":"34ba5c5227a6e1a158e9cb45b25f5cd83389402c767b8679f332ad172796aab1","abstract_canon_sha256":"18f038a926b6d00c2067f2e4b3148de47bc5196462e17e9a9de1548518d591ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:15.947130Z","signature_b64":"ND+Pd/q1sk57y3g00V+URgI8/10PqBCSdTiDzJ8Lc63vvvwPOZe5Si39CaKbbA1bfEnTfq3bgYQdNjoAi3PWBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f1b18da85befc6c145887f69015205667729e68e2ec51c95d2828b36369885e","last_reissued_at":"2026-05-18T00:01:15.946526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:15.946526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Hamilton cycles in Erd\\H{o}s-R\\'{e}nyi subgraphs of large graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tony Johansson","submitted_at":"2018-11-08T15:43:19Z","abstract_excerpt":"Given a graph $\\Gamma = (V, E)$ on $n$ vertices and $m$ edges, we define the Erd\\H{o}s-R\\'{e}nyi graph process with host $\\Gamma$ as follows. A permutation $e_1,\\dots,e_m$ of $E$ is chosen uniformly at random, and for $t\\leq m$ we let $\\Gamma_t = (V, \\{e_1,\\dots,e_t\\})$. Suppose the minimum degree of $\\Gamma$ is $\\delta(\\Gamma) \\geq (1/2 + \\varepsilon)n$ for some constant $\\varepsilon > 0$. Then with high probability, $\\Gamma_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two.\n  Given $0\\leq p\\leq 1$ we let $\\Gamma_p$ be the Erd\\H{o}s-R\\'{e}nyi subgraph of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.03501","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1811.03501","created_at":"2026-05-18T00:01:15.946621+00:00"},{"alias_kind":"arxiv_version","alias_value":"1811.03501v1","created_at":"2026-05-18T00:01:15.946621+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.03501","created_at":"2026-05-18T00:01:15.946621+00:00"},{"alias_kind":"pith_short_12","alias_value":"L4NRRWUFX36G","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_16","alias_value":"L4NRRWUFX36GYFCY","created_at":"2026-05-18T12:32:33.847187+00:00"},{"alias_kind":"pith_short_8","alias_value":"L4NRRWUF","created_at":"2026-05-18T12:32:33.847187+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ","json":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ.json","graph_json":"https://pith.science/api/pith-number/L4NRRWUFX36GYFCYQ73JAFJAKZ/graph.json","events_json":"https://pith.science/api/pith-number/L4NRRWUFX36GYFCYQ73JAFJAKZ/events.json","paper":"https://pith.science/paper/L4NRRWUF"},"agent_actions":{"view_html":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ","download_json":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ.json","view_paper":"https://pith.science/paper/L4NRRWUF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1811.03501&json=true","fetch_graph":"https://pith.science/api/pith-number/L4NRRWUFX36GYFCYQ73JAFJAKZ/graph.json","fetch_events":"https://pith.science/api/pith-number/L4NRRWUFX36GYFCYQ73JAFJAKZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ/action/storage_attestation","attest_author":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ/action/author_attestation","sign_citation":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ/action/citation_signature","submit_replication":"https://pith.science/pith/L4NRRWUFX36GYFCYQ73JAFJAKZ/action/replication_record"}},"created_at":"2026-05-18T00:01:15.946621+00:00","updated_at":"2026-05-18T00:01:15.946621+00:00"}