{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:ZPO6ZWDIG3ZEJUNZCMU46YB6L7","short_pith_number":"pith:ZPO6ZWDI","canonical_record":{"source":{"id":"0908.4572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-31T16:21:29Z","cross_cats_sorted":[],"title_canon_sha256":"01b80d63b4c2bfd12b705fea97a5fd91c9318583d56830ab25351849aca4ed17","abstract_canon_sha256":"4ab0d0d590289188973c07faa06aa55e5df79542fdf09d112433449453ecb588"},"schema_version":"1.0"},"canonical_sha256":"cbddecd86836f244d1b91329cf603e5fd1e48d6b56ca1e738881b00b92d45622","source":{"kind":"arxiv","id":"0908.4572","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.4572","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"0908.4572v2","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.4572","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"ZPO6ZWDIG3ZE","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZPO6ZWDIG3ZEJUNZ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZPO6ZWDI","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:ZPO6ZWDIG3ZEJUNZCMU46YB6L7","target":"record","payload":{"canonical_record":{"source":{"id":"0908.4572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-31T16:21:29Z","cross_cats_sorted":[],"title_canon_sha256":"01b80d63b4c2bfd12b705fea97a5fd91c9318583d56830ab25351849aca4ed17","abstract_canon_sha256":"4ab0d0d590289188973c07faa06aa55e5df79542fdf09d112433449453ecb588"},"schema_version":"1.0"},"canonical_sha256":"cbddecd86836f244d1b91329cf603e5fd1e48d6b56ca1e738881b00b92d45622","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:47.481968Z","signature_b64":"mGlJ2aOwJI7Tx8+JgdLmMa12AH1aB5p/iCT1l80pKjGZ9adEfjrKB0wQfZAtwgGIabYpa2BXc+Er016MdY/+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cbddecd86836f244d1b91329cf603e5fd1e48d6b56ca1e738881b00b92d45622","last_reissued_at":"2026-05-18T02:24:47.481367Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:47.481367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.4572","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":"2026-05-18T02:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WlGYYj8XxADhwPdDk0PhatNJrFL6gwDmmov7CUqD97yMpoQWi7gcjeK7fAU/dYaR90tM+rrxv8TiQu64jLR2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:40:47.274779Z"},"content_sha256":"f0b151538377911c22f7304572a642acd81dc371975a1722905a070a2ff8a004","schema_version":"1.0","event_id":"sha256:f0b151538377911c22f7304572a642acd81dc371975a1722905a070a2ff8a004"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:ZPO6ZWDIG3ZEJUNZCMU46YB6L7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Edge-disjoint Hamilton cycles in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniela K\\\"uhn, Demetres Christofides, Deryk Osthus","submitted_at":"2009-08-31T16:21:29Z","abstract_excerpt":"In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \\alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \\alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \\delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \\alpha > 0, every (almost) regular and sufficiently large "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.4572","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":""},"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":"2026-05-18T02:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/MKZhOZyS4H5RulVTMBe2kWUn5YryE44amichAEm/adaild0lXKdxupuIwl0RoQ85Q10zrsT+mvDp6I2COZXBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:40:47.275131Z"},"content_sha256":"ebcc0f45c632c8f2550c6b366c51742dff5febf4b9638001529efa2f98ddfcab","schema_version":"1.0","event_id":"sha256:ebcc0f45c632c8f2550c6b366c51742dff5febf4b9638001529efa2f98ddfcab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/bundle.json","state_url":"https://pith.science/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/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":"2026-05-27T05:40:47Z","links":{"resolver":"https://pith.science/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7","bundle":"https://pith.science/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/bundle.json","state":"https://pith.science/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZPO6ZWDIG3ZEJUNZCMU46YB6L7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:ZPO6ZWDIG3ZEJUNZCMU46YB6L7","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":"4ab0d0d590289188973c07faa06aa55e5df79542fdf09d112433449453ecb588","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-31T16:21:29Z","title_canon_sha256":"01b80d63b4c2bfd12b705fea97a5fd91c9318583d56830ab25351849aca4ed17"},"schema_version":"1.0","source":{"id":"0908.4572","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.4572","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"0908.4572v2","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.4572","created_at":"2026-05-18T02:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"ZPO6ZWDIG3ZE","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZPO6ZWDIG3ZEJUNZ","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZPO6ZWDI","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:ebcc0f45c632c8f2550c6b366c51742dff5febf4b9638001529efa2f98ddfcab","target":"graph","created_at":"2026-05-18T02:24:47Z","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":"In this paper we give an approximate answer to a question of Nash-Williams from 1970: we show that for every \\alpha > 0, every sufficiently large graph on n vertices with minimum degree at least (1/2 + \\alpha)n contains at least n/8 edge-disjoint Hamilton cycles. More generally, we give an asymptotically best possible answer for the number of edge-disjoint Hamilton cycles that a graph G with minimum degree \\delta must have. We also prove an approximate version of another long-standing conjecture of Nash-Williams: we show that for every \\alpha > 0, every (almost) regular and sufficiently large ","authors_text":"Daniela K\\\"uhn, Demetres Christofides, Deryk Osthus","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-31T16:21:29Z","title":"Edge-disjoint Hamilton cycles in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.4572","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:f0b151538377911c22f7304572a642acd81dc371975a1722905a070a2ff8a004","target":"record","created_at":"2026-05-18T02:24:47Z","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":"4ab0d0d590289188973c07faa06aa55e5df79542fdf09d112433449453ecb588","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-08-31T16:21:29Z","title_canon_sha256":"01b80d63b4c2bfd12b705fea97a5fd91c9318583d56830ab25351849aca4ed17"},"schema_version":"1.0","source":{"id":"0908.4572","kind":"arxiv","version":2}},"canonical_sha256":"cbddecd86836f244d1b91329cf603e5fd1e48d6b56ca1e738881b00b92d45622","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cbddecd86836f244d1b91329cf603e5fd1e48d6b56ca1e738881b00b92d45622","first_computed_at":"2026-05-18T02:24:47.481367Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:47.481367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mGlJ2aOwJI7Tx8+JgdLmMa12AH1aB5p/iCT1l80pKjGZ9adEfjrKB0wQfZAtwgGIabYpa2BXc+Er016MdY/+DA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:47.481968Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.4572","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0b151538377911c22f7304572a642acd81dc371975a1722905a070a2ff8a004","sha256:ebcc0f45c632c8f2550c6b366c51742dff5febf4b9638001529efa2f98ddfcab"],"state_sha256":"a6da8e50ec2c22af1750b499704168abaaef1f0a22416bf5784b38b24e694d1b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"01IoX9bty+GQ3pjcZhswoTN3obasYZ9g/AqNsEQF8zR8ueNkeGXukKIitNGRZrcWMT/+TI7EC+T3LFblGoEaBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:40:47.278148Z","bundle_sha256":"b2d7f55769f62ea239fa21beb8a9d0bb231c8a4ad8f8554a0ccfde0f6c13bb64"}}