{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:2LUXTYCUHGRZ5KG2BQHOJI24NP","short_pith_number":"pith:2LUXTYCU","schema_version":"1.0","canonical_sha256":"d2e979e05439a39ea8da0c0ee4a35c6bce81faed64ef598f218f9e356c7437fb","source":{"kind":"arxiv","id":"2303.11893","version":2},"attestation_state":"computed","paper":{"title":"Some results on the saturation number for unions of cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Chen, Xiying Yuan","submitted_at":"2023-03-21T14:36:39Z","abstract_excerpt":"Graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ and $H$ is a subgraph of $G+e$ for any edge $e$ not in $G$. The saturation number for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. In this paper, the saturation number for $K_p\\cup (t-1)K_q$ ($t\\geqslant 3$ and $2\\leqslant p<q$) is determined, and the extremal graph for $K_p\\cup 2K_q$ is determined. Moreover, the saturation number and the extremal graph for $K_p\\cup K_q\\cup K_r$ ($ r\\geqslant p+q$) are completely determined."},"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":"2303.11893","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-03-21T14:36:39Z","cross_cats_sorted":[],"title_canon_sha256":"d3b476d974389fd7290f65db24950fb745ecbbd274dff3e0ee0d1193b217cf4c","abstract_canon_sha256":"ca57259e728eb56ed01b93fb2f5a6ba3ef8e96968ef591fdd47c9565b5147e3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:58:55.648846Z","signature_b64":"4nDre+XtTcG8hz2GklOK1g+g9EwogQ+JP+EFZHUGBvUhE+jte0RCOq3Ei7+qJfjuuDHlCtq1nr7e1YobbpM1Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d2e979e05439a39ea8da0c0ee4a35c6bce81faed64ef598f218f9e356c7437fb","last_reissued_at":"2026-07-05T06:58:55.648333Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:58:55.648333Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some results on the saturation number for unions of cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Chen, Xiying Yuan","submitted_at":"2023-03-21T14:36:39Z","abstract_excerpt":"Graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ and $H$ is a subgraph of $G+e$ for any edge $e$ not in $G$. The saturation number for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. In this paper, the saturation number for $K_p\\cup (t-1)K_q$ ($t\\geqslant 3$ and $2\\leqslant p<q$) is determined, and the extremal graph for $K_p\\cup 2K_q$ is determined. Moreover, the saturation number and the extremal graph for $K_p\\cup K_q\\cup K_r$ ($ r\\geqslant p+q$) are completely determined."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.11893","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.11893/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2303.11893","created_at":"2026-07-05T06:58:55.648389+00:00"},{"alias_kind":"arxiv_version","alias_value":"2303.11893v2","created_at":"2026-07-05T06:58:55.648389+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.11893","created_at":"2026-07-05T06:58:55.648389+00:00"},{"alias_kind":"pith_short_12","alias_value":"2LUXTYCUHGRZ","created_at":"2026-07-05T06:58:55.648389+00:00"},{"alias_kind":"pith_short_16","alias_value":"2LUXTYCUHGRZ5KG2","created_at":"2026-07-05T06:58:55.648389+00:00"},{"alias_kind":"pith_short_8","alias_value":"2LUXTYCU","created_at":"2026-07-05T06:58:55.648389+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/2LUXTYCUHGRZ5KG2BQHOJI24NP","json":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP.json","graph_json":"https://pith.science/api/pith-number/2LUXTYCUHGRZ5KG2BQHOJI24NP/graph.json","events_json":"https://pith.science/api/pith-number/2LUXTYCUHGRZ5KG2BQHOJI24NP/events.json","paper":"https://pith.science/paper/2LUXTYCU"},"agent_actions":{"view_html":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP","download_json":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP.json","view_paper":"https://pith.science/paper/2LUXTYCU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2303.11893&json=true","fetch_graph":"https://pith.science/api/pith-number/2LUXTYCUHGRZ5KG2BQHOJI24NP/graph.json","fetch_events":"https://pith.science/api/pith-number/2LUXTYCUHGRZ5KG2BQHOJI24NP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP/action/storage_attestation","attest_author":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP/action/author_attestation","sign_citation":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP/action/citation_signature","submit_replication":"https://pith.science/pith/2LUXTYCUHGRZ5KG2BQHOJI24NP/action/replication_record"}},"created_at":"2026-07-05T06:58:55.648389+00:00","updated_at":"2026-07-05T06:58:55.648389+00:00"}