{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:GSQOPNSKQP6ZEUW4RCPJQB5XBM","short_pith_number":"pith:GSQOPNSK","canonical_record":{"source":{"id":"1201.6483","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T09:21:14Z","cross_cats_sorted":[],"title_canon_sha256":"3c00f542be8685e3e547af3d03402fbbb08d3854fb323f41eef5561003b6bd08","abstract_canon_sha256":"437e4dcd89552caaea089c70c11b71eece3fceeca9f265579434eef98e0a2139"},"schema_version":"1.0"},"canonical_sha256":"34a0e7b64a83fd9252dc889e9807b70b1966db8acc26bd6534a7bab6d0db2fb4","source":{"kind":"arxiv","id":"1201.6483","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.6483","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1201.6483v1","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6483","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"GSQOPNSKQP6Z","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GSQOPNSKQP6ZEUW4","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GSQOPNSK","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:GSQOPNSKQP6ZEUW4RCPJQB5XBM","target":"record","payload":{"canonical_record":{"source":{"id":"1201.6483","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T09:21:14Z","cross_cats_sorted":[],"title_canon_sha256":"3c00f542be8685e3e547af3d03402fbbb08d3854fb323f41eef5561003b6bd08","abstract_canon_sha256":"437e4dcd89552caaea089c70c11b71eece3fceeca9f265579434eef98e0a2139"},"schema_version":"1.0"},"canonical_sha256":"34a0e7b64a83fd9252dc889e9807b70b1966db8acc26bd6534a7bab6d0db2fb4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:31.396817Z","signature_b64":"jWKt4aTbNrHeUKrk2eg5+YTo5eBUfKt/7Wwl7YjbGMRjOD6oW+okHosyOFEMhhuRjNeTROytLM1xXAjRKiJWAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34a0e7b64a83fd9252dc889e9807b70b1966db8acc26bd6534a7bab6d0db2fb4","last_reissued_at":"2026-05-18T04:03:31.396133Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:31.396133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.6483","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":"2026-05-18T04:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZCWNNrXzhghArkTqC7Vyx/gFwx2NiTRY9SXW/qfKfMuC/X/BjBp7hrJOo3untLVNEDCiKPHVFRw9kn9kmBCAAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T19:45:02.483481Z"},"content_sha256":"3c67d30fc2b00bd25b191d9c8f7d63078617de651e27b3409e238171bd4ea814","schema_version":"1.0","event_id":"sha256:3c67d30fc2b00bd25b191d9c8f7d63078617de651e27b3409e238171bd4ea814"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:GSQOPNSKQP6ZEUW4RCPJQB5XBM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The thickness of amalgamations of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiangheng Kong, Yan Yang","submitted_at":"2012-01-31T09:21:14Z","abstract_excerpt":"The thickness $\\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. As a topological invariant of a graph, it is a measurement of the closeness to planarity of a graph, and it also has important applications to VLSI design. In this paper, the thickness of graphs that are obtained by vertex-amalgamation and bar-amalgamation of any two graphs whose thicknesses are known are obtained, respectively. And the lower and upper bounds for the thickness of graphs that are obtained by edge-amalgamation and 2-vertex-amalgamation of any tw"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6483","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":"2026-05-18T04:03:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FKOHUgDYG0ZJ5KV9NdAB1rSxTEFyjeJRba5td/nokjgRxcO1RVM3/XM3vhCeIRg0sbkGFPzGLGdv04JUdZBIAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-18T19:45:02.484032Z"},"content_sha256":"d3cc089791dc235abfb872400def3ac41ef947f7daa1d3f869d0b91803892da8","schema_version":"1.0","event_id":"sha256:d3cc089791dc235abfb872400def3ac41ef947f7daa1d3f869d0b91803892da8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/bundle.json","state_url":"https://pith.science/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/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":"2026-05-18T19:45:02Z","links":{"resolver":"https://pith.science/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM","bundle":"https://pith.science/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/bundle.json","state":"https://pith.science/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GSQOPNSKQP6ZEUW4RCPJQB5XBM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:GSQOPNSKQP6ZEUW4RCPJQB5XBM","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":"437e4dcd89552caaea089c70c11b71eece3fceeca9f265579434eef98e0a2139","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T09:21:14Z","title_canon_sha256":"3c00f542be8685e3e547af3d03402fbbb08d3854fb323f41eef5561003b6bd08"},"schema_version":"1.0","source":{"id":"1201.6483","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.6483","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1201.6483v1","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6483","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"GSQOPNSKQP6Z","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"GSQOPNSKQP6ZEUW4","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"GSQOPNSK","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:d3cc089791dc235abfb872400def3ac41ef947f7daa1d3f869d0b91803892da8","target":"graph","created_at":"2026-05-18T04:03:31Z","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 thickness $\\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. As a topological invariant of a graph, it is a measurement of the closeness to planarity of a graph, and it also has important applications to VLSI design. In this paper, the thickness of graphs that are obtained by vertex-amalgamation and bar-amalgamation of any two graphs whose thicknesses are known are obtained, respectively. And the lower and upper bounds for the thickness of graphs that are obtained by edge-amalgamation and 2-vertex-amalgamation of any tw","authors_text":"Xiangheng Kong, Yan Yang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T09:21:14Z","title":"The thickness of amalgamations of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6483","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":"sha256:3c67d30fc2b00bd25b191d9c8f7d63078617de651e27b3409e238171bd4ea814","target":"record","created_at":"2026-05-18T04:03:31Z","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":"437e4dcd89552caaea089c70c11b71eece3fceeca9f265579434eef98e0a2139","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-31T09:21:14Z","title_canon_sha256":"3c00f542be8685e3e547af3d03402fbbb08d3854fb323f41eef5561003b6bd08"},"schema_version":"1.0","source":{"id":"1201.6483","kind":"arxiv","version":1}},"canonical_sha256":"34a0e7b64a83fd9252dc889e9807b70b1966db8acc26bd6534a7bab6d0db2fb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"34a0e7b64a83fd9252dc889e9807b70b1966db8acc26bd6534a7bab6d0db2fb4","first_computed_at":"2026-05-18T04:03:31.396133Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:31.396133Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jWKt4aTbNrHeUKrk2eg5+YTo5eBUfKt/7Wwl7YjbGMRjOD6oW+okHosyOFEMhhuRjNeTROytLM1xXAjRKiJWAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:31.396817Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.6483","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c67d30fc2b00bd25b191d9c8f7d63078617de651e27b3409e238171bd4ea814","sha256:d3cc089791dc235abfb872400def3ac41ef947f7daa1d3f869d0b91803892da8"],"state_sha256":"cc25920a2c71f0764f8a474262b83b3318a1d92f98a52778f612455f04d5d999"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rHDUTqUT4girFzezVRW7WP1trJCvs0C5WjV6xwfLHgPIoIZGHMg8dRCdxHF6z6jImfNXhsoxRWaomxl0YeDpAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-18T19:45:02.485876Z","bundle_sha256":"eb8125c25e8969b7123b86fe32461ff6267d8faf5174e6799bcd854954048765"}}