{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:3RR3B6YWDX6YVHLCDN54PS63ER","short_pith_number":"pith:3RR3B6YW","schema_version":"1.0","canonical_sha256":"dc63b0fb161dfd8a9d621b7bc7cbdb245eba696993fe590d1e88be5ba8a05a36","source":{"kind":"arxiv","id":"1901.08052","version":1},"attestation_state":"computed","paper":{"title":"The thickness of the Kronecker product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xia Guo, Yan Yang","submitted_at":"2019-01-23T02:00:30Z","abstract_excerpt":"The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. In this paper, we present sharp lower and upper bounds for the thickness of the Kronecker product $G\\times H$ of two graphs $G$ and $H$. We also give the exact thickness numbers for the Kronecker product graphs $K_n\\times K_2$, $K_{m,n}\\times K_2$ and $K_{n,n,n}\\times K_2$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.08052","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-23T02:00:30Z","cross_cats_sorted":[],"title_canon_sha256":"ec64493b1c61a3ab5db3d5359a8ca500870ce396c9d5d0421d411a1090c197ba","abstract_canon_sha256":"68325b72e192a04fa9e60b8618ea85bbb921244915cfe4709c941aded8222976"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:37.992499Z","signature_b64":"dkmsvfXE6pZzBrXqYvgddp6EJz0qvYtCmFr6L3GLdCP+OnpiuejQgADlCr2TsQlkmfwzGK8y57eRTJ/K3HBeCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc63b0fb161dfd8a9d621b7bc7cbdb245eba696993fe590d1e88be5ba8a05a36","last_reissued_at":"2026-05-17T23:55:37.992062Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:37.992062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The thickness of the Kronecker product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xia Guo, Yan Yang","submitted_at":"2019-01-23T02:00:30Z","abstract_excerpt":"The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. In this paper, we present sharp lower and upper bounds for the thickness of the Kronecker product $G\\times H$ of two graphs $G$ and $H$. We also give the exact thickness numbers for the Kronecker product graphs $K_n\\times K_2$, $K_{m,n}\\times K_2$ and $K_{n,n,n}\\times K_2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.08052","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.08052","created_at":"2026-05-17T23:55:37.992127+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.08052v1","created_at":"2026-05-17T23:55:37.992127+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.08052","created_at":"2026-05-17T23:55:37.992127+00:00"},{"alias_kind":"pith_short_12","alias_value":"3RR3B6YWDX6Y","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"3RR3B6YWDX6YVHLC","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"3RR3B6YW","created_at":"2026-05-18T12:33:10.108867+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER","json":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER.json","graph_json":"https://pith.science/api/pith-number/3RR3B6YWDX6YVHLCDN54PS63ER/graph.json","events_json":"https://pith.science/api/pith-number/3RR3B6YWDX6YVHLCDN54PS63ER/events.json","paper":"https://pith.science/paper/3RR3B6YW"},"agent_actions":{"view_html":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER","download_json":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER.json","view_paper":"https://pith.science/paper/3RR3B6YW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.08052&json=true","fetch_graph":"https://pith.science/api/pith-number/3RR3B6YWDX6YVHLCDN54PS63ER/graph.json","fetch_events":"https://pith.science/api/pith-number/3RR3B6YWDX6YVHLCDN54PS63ER/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER/action/storage_attestation","attest_author":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER/action/author_attestation","sign_citation":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER/action/citation_signature","submit_replication":"https://pith.science/pith/3RR3B6YWDX6YVHLCDN54PS63ER/action/replication_record"}},"created_at":"2026-05-17T23:55:37.992127+00:00","updated_at":"2026-05-17T23:55:37.992127+00:00"}