{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PNLLKCBUOFKKK5MJTPK72C6CIT","short_pith_number":"pith:PNLLKCBU","canonical_record":{"source":{"id":"1606.03953","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-13T13:59:25Z","cross_cats_sorted":[],"title_canon_sha256":"e1307453037d9833a9042925783aec0fbe18f8d57e88c23361ba7a68fc19d2af","abstract_canon_sha256":"4eb9e74fc00e95274d17729dea1c643ede59b218f78c265850ecee0c0c4ef82d"},"schema_version":"1.0"},"canonical_sha256":"7b56b508347154a575899bd5fd0bc244f59e3ab9b65f63c27023d701857fdbb4","source":{"kind":"arxiv","id":"1606.03953","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.03953","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"arxiv_version","alias_value":"1606.03953v2","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03953","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"pith_short_12","alias_value":"PNLLKCBUOFKK","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PNLLKCBUOFKKK5MJ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PNLLKCBU","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PNLLKCBUOFKKK5MJTPK72C6CIT","target":"record","payload":{"canonical_record":{"source":{"id":"1606.03953","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-13T13:59:25Z","cross_cats_sorted":[],"title_canon_sha256":"e1307453037d9833a9042925783aec0fbe18f8d57e88c23361ba7a68fc19d2af","abstract_canon_sha256":"4eb9e74fc00e95274d17729dea1c643ede59b218f78c265850ecee0c0c4ef82d"},"schema_version":"1.0"},"canonical_sha256":"7b56b508347154a575899bd5fd0bc244f59e3ab9b65f63c27023d701857fdbb4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:26.297071Z","signature_b64":"q2rRWgI97Cq1I9jf90WmPvw/0Pr+Y96Ub2a07Mp4jZTrXBDr/ydpiXnUGg11+23yr8T96dC2CQ6DvIjgcnZCAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b56b508347154a575899bd5fd0bc244f59e3ab9b65f63c27023d701857fdbb4","last_reissued_at":"2026-05-17T23:51:26.296585Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:26.296585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.03953","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":"2026-05-17T23:51:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zaMhTniw//X/HCYnjjYOa/l2SG79Rp44MqWgcTvd1YNna0OpCLZ9ySkpeHsuvG42cI230pq5TTlSbgTt6VDfBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:28:40.412785Z"},"content_sha256":"0f7d77dc2f47ed49a4d068afd4cf56c27a51906386d5eb9f247426595d925492","schema_version":"1.0","event_id":"sha256:0f7d77dc2f47ed49a4d068afd4cf56c27a51906386d5eb9f247426595d925492"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PNLLKCBUOFKKK5MJTPK72C6CIT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal packings of bounded degree trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniela K\\\"uhn, Deryk Osthus, Felix Joos, Jaehoon Kim","submitted_at":"2016-06-13T13:59:25Z","abstract_excerpt":"We prove that if $T_1,\\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\\dots, T_n$. This shows that the tree packing conjecture of Gy\\'arf\\'as and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03953","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":"2026-05-17T23:51:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7v2jcg6kiXLtgT6P62V2KSDCUSWnV+XJdihQg7YDkmcEwCr5oVOrz/bC0wz0ue/b1f1EDZIjCACME29pzIPgAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:28:40.413417Z"},"content_sha256":"82f18d32f5b892cf9ab8dbe14899e395df909b281e94f85a35df8cb82b5c4f61","schema_version":"1.0","event_id":"sha256:82f18d32f5b892cf9ab8dbe14899e395df909b281e94f85a35df8cb82b5c4f61"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/bundle.json","state_url":"https://pith.science/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/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":"2026-05-27T08:28:40Z","links":{"resolver":"https://pith.science/pith/PNLLKCBUOFKKK5MJTPK72C6CIT","bundle":"https://pith.science/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/bundle.json","state":"https://pith.science/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PNLLKCBUOFKKK5MJTPK72C6CIT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PNLLKCBUOFKKK5MJTPK72C6CIT","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":"4eb9e74fc00e95274d17729dea1c643ede59b218f78c265850ecee0c0c4ef82d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-13T13:59:25Z","title_canon_sha256":"e1307453037d9833a9042925783aec0fbe18f8d57e88c23361ba7a68fc19d2af"},"schema_version":"1.0","source":{"id":"1606.03953","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.03953","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"arxiv_version","alias_value":"1606.03953v2","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03953","created_at":"2026-05-17T23:51:26Z"},{"alias_kind":"pith_short_12","alias_value":"PNLLKCBUOFKK","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PNLLKCBUOFKKK5MJ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PNLLKCBU","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:82f18d32f5b892cf9ab8dbe14899e395df909b281e94f85a35df8cb82b5c4f61","target":"graph","created_at":"2026-05-17T23:51:26Z","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":"We prove that if $T_1,\\dots, T_n$ is a sequence of bounded degree trees so that $T_i$ has $i$ vertices, then $K_n$ has a decomposition into $T_1,\\dots, T_n$. This shows that the tree packing conjecture of Gy\\'arf\\'as and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first $o(n)$ trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our p","authors_text":"Daniela K\\\"uhn, Deryk Osthus, Felix Joos, Jaehoon Kim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-13T13:59:25Z","title":"Optimal packings of bounded degree trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03953","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":"sha256:0f7d77dc2f47ed49a4d068afd4cf56c27a51906386d5eb9f247426595d925492","target":"record","created_at":"2026-05-17T23:51:26Z","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":"4eb9e74fc00e95274d17729dea1c643ede59b218f78c265850ecee0c0c4ef82d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-13T13:59:25Z","title_canon_sha256":"e1307453037d9833a9042925783aec0fbe18f8d57e88c23361ba7a68fc19d2af"},"schema_version":"1.0","source":{"id":"1606.03953","kind":"arxiv","version":2}},"canonical_sha256":"7b56b508347154a575899bd5fd0bc244f59e3ab9b65f63c27023d701857fdbb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b56b508347154a575899bd5fd0bc244f59e3ab9b65f63c27023d701857fdbb4","first_computed_at":"2026-05-17T23:51:26.296585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:26.296585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"q2rRWgI97Cq1I9jf90WmPvw/0Pr+Y96Ub2a07Mp4jZTrXBDr/ydpiXnUGg11+23yr8T96dC2CQ6DvIjgcnZCAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:26.297071Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.03953","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f7d77dc2f47ed49a4d068afd4cf56c27a51906386d5eb9f247426595d925492","sha256:82f18d32f5b892cf9ab8dbe14899e395df909b281e94f85a35df8cb82b5c4f61"],"state_sha256":"fef3adac865550ea64d6a7383ac0921a455b1228687d7050c2c1706db9736d00"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O0teab/cvfZFB3TbZaoOOqnUuGfMQJnT+7iuPw/dGYqLp6grniDPoIkjiuLr3WBYWYettDrAdFCxakpkiWbYBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T08:28:40.416641Z","bundle_sha256":"c44c689a1f823aad577f2a3356bf767a6ddb0ea38fbd7c1cc750a93a8401e466"}}