{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:RZLCBRKPCXQ4IWR6MH4GAQY5WX","short_pith_number":"pith:RZLCBRKP","canonical_record":{"source":{"id":"1211.4437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-19T14:36:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d1921453f635f8768c65c0bfda2042aa3b25dfd5412292e1ee4e520f92b8b76d","abstract_canon_sha256":"df588a4152f627f4a5e44a62b7f8f7fad111b77e984d810dfe2547e819a8e2c2"},"schema_version":"1.0"},"canonical_sha256":"8e5620c54f15e1c45a3e61f860431db5ecbcc38b1be7a7597145875160987b01","source":{"kind":"arxiv","id":"1211.4437","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.4437","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"arxiv_version","alias_value":"1211.4437v1","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4437","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"pith_short_12","alias_value":"RZLCBRKPCXQ4","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RZLCBRKPCXQ4IWR6","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RZLCBRKP","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:RZLCBRKPCXQ4IWR6MH4GAQY5WX","target":"record","payload":{"canonical_record":{"source":{"id":"1211.4437","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-19T14:36:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d1921453f635f8768c65c0bfda2042aa3b25dfd5412292e1ee4e520f92b8b76d","abstract_canon_sha256":"df588a4152f627f4a5e44a62b7f8f7fad111b77e984d810dfe2547e819a8e2c2"},"schema_version":"1.0"},"canonical_sha256":"8e5620c54f15e1c45a3e61f860431db5ecbcc38b1be7a7597145875160987b01","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:26.932381Z","signature_b64":"+lutRFE279wanYs6KCavRSAKhBYN5DwjwQAn/7fDo7pFJ6fSOteL+SeKpuaWN9YPGxzVVIrDE5dGt96yy4QIAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e5620c54f15e1c45a3e61f860431db5ecbcc38b1be7a7597145875160987b01","last_reissued_at":"2026-05-18T03:40:26.931641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:26.931641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.4437","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:40:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qvV3W9TxgTrkGW9s+kLVNuY7Tn/AXARu1wYv3yQdwJsA9gwXrZBKHvGqhfAL40nsFPV6OtJh28LH1MTrm2tCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:22:30.392907Z"},"content_sha256":"496077291500fc178eee61700ec1479b7f879b02d8655fa1d5a9ee170492a59d","schema_version":"1.0","event_id":"sha256:496077291500fc178eee61700ec1479b7f879b02d8655fa1d5a9ee170492a59d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:RZLCBRKPCXQ4IWR6MH4GAQY5WX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The crossing numbers of $K_{n,n}-nK_2$, $K_{n}\\times P_2$, $K_{n}\\times P_3$ and $K_n\\times C_4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Baigong Zheng, Xiaohui Lin, Xirong Xu, Yuansheng Yang","submitted_at":"2012-11-19T14:36:17Z","abstract_excerpt":"The crossing number of a graph $G$ is the minimum number of pairwise intersections of edges among all drawings of $G$. In this paper, we study the crossing number of $K_{n,n}-nK_2$, $K_n\\times P_2$, $K_n\\times P_3$ and $K_n\\times C_4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4437","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:40:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n+p6OS58giV1zI1YYfQH4y6bk+hxL7kruNQFEk8JD4XHraK5pvkgtRKnHiaTI3aK3gnNSoR3iykuVqmLvSgiAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T22:22:30.393240Z"},"content_sha256":"bd4a2500bad8cf5ebc1f527bb22f94c023eae23f4afacdd4eb49d6668f518c9d","schema_version":"1.0","event_id":"sha256:bd4a2500bad8cf5ebc1f527bb22f94c023eae23f4afacdd4eb49d6668f518c9d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/bundle.json","state_url":"https://pith.science/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/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":"2026-06-01T22:22:30Z","links":{"resolver":"https://pith.science/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX","bundle":"https://pith.science/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/bundle.json","state":"https://pith.science/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RZLCBRKPCXQ4IWR6MH4GAQY5WX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RZLCBRKPCXQ4IWR6MH4GAQY5WX","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":"df588a4152f627f4a5e44a62b7f8f7fad111b77e984d810dfe2547e819a8e2c2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-19T14:36:17Z","title_canon_sha256":"d1921453f635f8768c65c0bfda2042aa3b25dfd5412292e1ee4e520f92b8b76d"},"schema_version":"1.0","source":{"id":"1211.4437","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.4437","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"arxiv_version","alias_value":"1211.4437v1","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.4437","created_at":"2026-05-18T03:40:26Z"},{"alias_kind":"pith_short_12","alias_value":"RZLCBRKPCXQ4","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RZLCBRKPCXQ4IWR6","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RZLCBRKP","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:bd4a2500bad8cf5ebc1f527bb22f94c023eae23f4afacdd4eb49d6668f518c9d","target":"graph","created_at":"2026-05-18T03:40:26Z","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":"The crossing number of a graph $G$ is the minimum number of pairwise intersections of edges among all drawings of $G$. In this paper, we study the crossing number of $K_{n,n}-nK_2$, $K_n\\times P_2$, $K_n\\times P_3$ and $K_n\\times C_4$.","authors_text":"Baigong Zheng, Xiaohui Lin, Xirong Xu, Yuansheng Yang","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-19T14:36:17Z","title":"The crossing numbers of $K_{n,n}-nK_2$, $K_{n}\\times P_2$, $K_{n}\\times P_3$ and $K_n\\times C_4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4437","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":"sha256:496077291500fc178eee61700ec1479b7f879b02d8655fa1d5a9ee170492a59d","target":"record","created_at":"2026-05-18T03:40:26Z","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":"df588a4152f627f4a5e44a62b7f8f7fad111b77e984d810dfe2547e819a8e2c2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-11-19T14:36:17Z","title_canon_sha256":"d1921453f635f8768c65c0bfda2042aa3b25dfd5412292e1ee4e520f92b8b76d"},"schema_version":"1.0","source":{"id":"1211.4437","kind":"arxiv","version":1}},"canonical_sha256":"8e5620c54f15e1c45a3e61f860431db5ecbcc38b1be7a7597145875160987b01","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e5620c54f15e1c45a3e61f860431db5ecbcc38b1be7a7597145875160987b01","first_computed_at":"2026-05-18T03:40:26.931641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:26.931641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+lutRFE279wanYs6KCavRSAKhBYN5DwjwQAn/7fDo7pFJ6fSOteL+SeKpuaWN9YPGxzVVIrDE5dGt96yy4QIAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:26.932381Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.4437","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:496077291500fc178eee61700ec1479b7f879b02d8655fa1d5a9ee170492a59d","sha256:bd4a2500bad8cf5ebc1f527bb22f94c023eae23f4afacdd4eb49d6668f518c9d"],"state_sha256":"c421d5a3db7517839f3aeebd52c0aa5311e72d9a80c06a4d1898ef20f27fceee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yxWaaMksfzohm0ObIhLP2oEF+JXBb19llfdivfhcy7B9Vs1/useceHY1N+vlLfZZ7hhU3z/VmP2eLdc5S9JsAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T22:22:30.395138Z","bundle_sha256":"36ac0e9376ed4300bbfe388a3f05657e198f5d9dc1c9228a92f1e99bae1a1344"}}