{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:X7QHZZGEPII3IR5HS4XHL22JX3","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":"104a1f57bbb6c4afada0f2e4622ed9b12634ca8f9e28fa6d486fd3fa1005ffa7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-28T08:06:38Z","title_canon_sha256":"026311f87d0ddaf0d20af1b6cc86fa11f3fa477c75f40d33feb0783e995478d8"},"schema_version":"1.0","source":{"id":"2605.29553","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.29553","created_at":"2026-05-29T01:05:46Z"},{"alias_kind":"arxiv_version","alias_value":"2605.29553v1","created_at":"2026-05-29T01:05:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.29553","created_at":"2026-05-29T01:05:46Z"},{"alias_kind":"pith_short_12","alias_value":"X7QHZZGEPII3","created_at":"2026-05-29T01:05:46Z"},{"alias_kind":"pith_short_16","alias_value":"X7QHZZGEPII3IR5H","created_at":"2026-05-29T01:05:46Z"},{"alias_kind":"pith_short_8","alias_value":"X7QHZZGE","created_at":"2026-05-29T01:05:46Z"}],"graph_snapshots":[{"event_id":"sha256:62f8d311c29569f2399713faadf466392db3f0f3e4092f5dfd72056e9fc7e908","target":"graph","created_at":"2026-05-29T01:05:46Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.29553/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\\alpha=\\alpha(n)=o(1)$, let $G_{\\alpha}$ be an $n$-vertex graph with minimum degree $\\delta(G_{\\alpha})\\ge\\alpha n$. We prove that if $$p\\ge(1+\\varepsilon)\\frac{\\log(1/\\alpha)}{n},$$ then the union $G_{\\alpha}\\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly strengthens a recent result of Hahn-Klimroth, Maesaka, Mogge, Mohr, and Parczyk by improving the leading constant from 6 to the optimal value of 1. Crucially, we show that this bound on $p$ is best possible when $\\a","authors_text":"Guorui Ma, Zhifei Yan","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-28T08:06:38Z","title":"Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.29553","kind":"arxiv","version":1},"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:b409ea4b97697d1e92d5bf70c634ff64419eb330f38c0786cb3c0cfb581baaff","target":"record","created_at":"2026-05-29T01:05:46Z","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":"104a1f57bbb6c4afada0f2e4622ed9b12634ca8f9e28fa6d486fd3fa1005ffa7","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-28T08:06:38Z","title_canon_sha256":"026311f87d0ddaf0d20af1b6cc86fa11f3fa477c75f40d33feb0783e995478d8"},"schema_version":"1.0","source":{"id":"2605.29553","kind":"arxiv","version":1}},"canonical_sha256":"bfe07ce4c47a11b447a7972e75eb49bedf31d4308bf090612df7d693303349a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfe07ce4c47a11b447a7972e75eb49bedf31d4308bf090612df7d693303349a0","first_computed_at":"2026-05-29T01:05:46.920473Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T01:05:46.920473Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uXMLZh6cFZ1DX1vi6stofiekAn2/yjG3D7bj0V6yJ2OlbQ3cQ8YKGn4+tVoFwchqVK1+e3dGIrn6cAhZlCi4AQ==","signature_status":"signed_v1","signed_at":"2026-05-29T01:05:46.921202Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.29553","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b409ea4b97697d1e92d5bf70c634ff64419eb330f38c0786cb3c0cfb581baaff","sha256:62f8d311c29569f2399713faadf466392db3f0f3e4092f5dfd72056e9fc7e908"],"state_sha256":"ea51121306b050148adf492e6dc43085762dfed1a463efacdcebcf9fe14cc7a6"}