{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:OBEZ2OKI2QLA7X4SQFCG4PVSXV","short_pith_number":"pith:OBEZ2OKI","canonical_record":{"source":{"id":"2007.02891","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-07-06T17:07:05Z","cross_cats_sorted":[],"title_canon_sha256":"a24ac0f1f668422c2c0c18167cce6867d94c3613883fd1888ba35c66f359342b","abstract_canon_sha256":"e0e7535f0c13bf4ada4f59795b5e8bf3de847570cad57b95a4a8ecb20c4d4b90"},"schema_version":"1.0"},"canonical_sha256":"70499d3948d4160fdf9281446e3eb2bd446ceccd09f781b36391df959f95a7ab","source":{"kind":"arxiv","id":"2007.02891","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.02891","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"arxiv_version","alias_value":"2007.02891v2","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.02891","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_12","alias_value":"OBEZ2OKI2QLA","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_16","alias_value":"OBEZ2OKI2QLA7X4S","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_8","alias_value":"OBEZ2OKI","created_at":"2026-07-05T04:48:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:OBEZ2OKI2QLA7X4SQFCG4PVSXV","target":"record","payload":{"canonical_record":{"source":{"id":"2007.02891","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-07-06T17:07:05Z","cross_cats_sorted":[],"title_canon_sha256":"a24ac0f1f668422c2c0c18167cce6867d94c3613883fd1888ba35c66f359342b","abstract_canon_sha256":"e0e7535f0c13bf4ada4f59795b5e8bf3de847570cad57b95a4a8ecb20c4d4b90"},"schema_version":"1.0"},"canonical_sha256":"70499d3948d4160fdf9281446e3eb2bd446ceccd09f781b36391df959f95a7ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:48:05.910479Z","signature_b64":"xXCzRxwH9ZJNmMy3k98hPoU7ToxcbULjRtixNdY3vKy6J+jTCtKr4iaC7MZ/wqEcIHUhGWadvmBnFkWanF0ABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70499d3948d4160fdf9281446e3eb2bd446ceccd09f781b36391df959f95a7ab","last_reissued_at":"2026-07-05T04:48:05.910072Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:48:05.910072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.02891","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:48:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dQiZ4QNUrBBi3UNHRR9TmwJgPpRWQR8AXUIgehgUt1YzCeJ8VZUG9ZJ9OA95i1NjiUmNjgWyhaBW9IdJjJfvAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T16:06:05.194626Z"},"content_sha256":"b41ddbdbe6c40de980932cb99a12e67c0e12c67f2bc5cafbc4d10626e9199871","schema_version":"1.0","event_id":"sha256:b41ddbdbe6c40de980932cb99a12e67c0e12c67f2bc5cafbc4d10626e9199871"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:OBEZ2OKI2QLA7X4SQFCG4PVSXV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hamiltonicity of random subgraphs of the hypercube","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alberto Espuny D\\'iaz, Ant\\'onio Gir\\~ao, Daniela K\\\"uhn, Deryk Osthus, Padraig Condon","submitted_at":"2020-07-06T17:07:05Z","abstract_excerpt":"We study Hamiltonicity in random subgraphs of the hypercube $\\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\\in\\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\\subseteq\\mathcal{Q}^n$ satisfies $\\delta(H)\\geq\\alpha n$ with $\\alpha>0$ fixed and we consider a random binomial sub"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.02891","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.02891/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:48:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"okLDaZILGMJw7jdl8C0aBWjCo5nMzdNUEY+PwITw+nvA1AjHyMhejp7m4cPv006FeVP/uilK6ljv5gpV0Qb1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T16:06:05.195157Z"},"content_sha256":"809bf5ef330aa30a2a89b0f21250e77eecdd1a75ad30e4986b96f8c2e9531ac0","schema_version":"1.0","event_id":"sha256:809bf5ef330aa30a2a89b0f21250e77eecdd1a75ad30e4986b96f8c2e9531ac0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/bundle.json","state_url":"https://pith.science/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-15T16:06:05Z","links":{"resolver":"https://pith.science/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV","bundle":"https://pith.science/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/bundle.json","state":"https://pith.science/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OBEZ2OKI2QLA7X4SQFCG4PVSXV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:OBEZ2OKI2QLA7X4SQFCG4PVSXV","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":"e0e7535f0c13bf4ada4f59795b5e8bf3de847570cad57b95a4a8ecb20c4d4b90","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-07-06T17:07:05Z","title_canon_sha256":"a24ac0f1f668422c2c0c18167cce6867d94c3613883fd1888ba35c66f359342b"},"schema_version":"1.0","source":{"id":"2007.02891","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.02891","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"arxiv_version","alias_value":"2007.02891v2","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.02891","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_12","alias_value":"OBEZ2OKI2QLA","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_16","alias_value":"OBEZ2OKI2QLA7X4S","created_at":"2026-07-05T04:48:05Z"},{"alias_kind":"pith_short_8","alias_value":"OBEZ2OKI","created_at":"2026-07-05T04:48:05Z"}],"graph_snapshots":[{"event_id":"sha256:809bf5ef330aa30a2a89b0f21250e77eecdd1a75ad30e4986b96f8c2e9531ac0","target":"graph","created_at":"2026-07-05T04:48:05Z","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/2007.02891/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study Hamiltonicity in random subgraphs of the hypercube $\\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\\mathcal{Q}^n$ according to a uniformly chosen random ordering. Then, with high probability, as soon as the graph produced by this process has minimum degree $2k$, it contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\\in\\mathbb{N}$. Secondly, we obtain a perturbation result: if $H\\subseteq\\mathcal{Q}^n$ satisfies $\\delta(H)\\geq\\alpha n$ with $\\alpha>0$ fixed and we consider a random binomial sub","authors_text":"Alberto Espuny D\\'iaz, Ant\\'onio Gir\\~ao, Daniela K\\\"uhn, Deryk Osthus, Padraig Condon","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-07-06T17:07:05Z","title":"Hamiltonicity of random subgraphs of the hypercube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.02891","kind":"arxiv","version":2},"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:b41ddbdbe6c40de980932cb99a12e67c0e12c67f2bc5cafbc4d10626e9199871","target":"record","created_at":"2026-07-05T04:48:05Z","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":"e0e7535f0c13bf4ada4f59795b5e8bf3de847570cad57b95a4a8ecb20c4d4b90","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2020-07-06T17:07:05Z","title_canon_sha256":"a24ac0f1f668422c2c0c18167cce6867d94c3613883fd1888ba35c66f359342b"},"schema_version":"1.0","source":{"id":"2007.02891","kind":"arxiv","version":2}},"canonical_sha256":"70499d3948d4160fdf9281446e3eb2bd446ceccd09f781b36391df959f95a7ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70499d3948d4160fdf9281446e3eb2bd446ceccd09f781b36391df959f95a7ab","first_computed_at":"2026-07-05T04:48:05.910072Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:48:05.910072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xXCzRxwH9ZJNmMy3k98hPoU7ToxcbULjRtixNdY3vKy6J+jTCtKr4iaC7MZ/wqEcIHUhGWadvmBnFkWanF0ABg==","signature_status":"signed_v1","signed_at":"2026-07-05T04:48:05.910479Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.02891","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b41ddbdbe6c40de980932cb99a12e67c0e12c67f2bc5cafbc4d10626e9199871","sha256:809bf5ef330aa30a2a89b0f21250e77eecdd1a75ad30e4986b96f8c2e9531ac0"],"state_sha256":"d1513bbf38f7634ec2ce4a2f94ec8baf4a1ffca3f3f2e10509ffb05e31d194e4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iApzAfXBeIOxxTModGABAiFGlsVyKGys0OX6qqWjIYsF82/DMtTpOjYGb/G6XwNJ4uanNkbf4p7IInMTyWtPDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T16:06:05.200265Z","bundle_sha256":"93c74e5f0f13aff0509519aa096a394d696c74c68f0614da58df46975a1a9dca"}}