{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:AIQSDWK5WXXZZDQD6QZAFFPKEA","short_pith_number":"pith:AIQSDWK5","schema_version":"1.0","canonical_sha256":"022121d95db5ef9c8e03f4320295ea202039af2194370007e93498caf6c076e0","source":{"kind":"arxiv","id":"1408.3137","version":1},"attestation_state":"computed","paper":{"title":"Graph Saturation in Multipartite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Florian Pfender, Michael Ferrara, Michael S. Jacobson, Paul S. Wenger","submitted_at":"2014-08-13T20:31:10Z","abstract_excerpt":"Let $G$ be a fixed graph and let ${\\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\\mathcal F}$-saturated if no member of ${\\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\\mathcal F}$ is a subgraph of $J+e$. We let $\\text{ex}({\\mathcal F},G)$ and $\\text{sat}({\\mathcal F},G)$ denote the maximum and minimum size of an ${\\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\\mathcal F}$ is a subgraph of $G$, then $\\text{sat}({\\mathcal F},G) = \\text{ex}({\\mathcal F}, G) = |E(G)|$.\n  In this paper, for $k\\ge 3$ and $n\\ge 1"},"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":"1408.3137","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-13T20:31:10Z","cross_cats_sorted":[],"title_canon_sha256":"436d19802bdc7afc6e1124a342315f9630c7b614dfe1b302ccaf477639509168","abstract_canon_sha256":"b87c10902aa43feb411238512ec693f6e6a2e387ad1c8f842f330d0b13d521f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:13.718520Z","signature_b64":"bfwyY6Db6TsJ2VRUceqyEBDXBiOSNFWKCTmx+BCicAh/LS+sUfzVM5t4uHovvwTh5MI/5/Bfb2mHsbqX7AfHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"022121d95db5ef9c8e03f4320295ea202039af2194370007e93498caf6c076e0","last_reissued_at":"2026-05-18T02:45:13.717864Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:13.717864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Graph Saturation in Multipartite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Florian Pfender, Michael Ferrara, Michael S. Jacobson, Paul S. Wenger","submitted_at":"2014-08-13T20:31:10Z","abstract_excerpt":"Let $G$ be a fixed graph and let ${\\mathcal F}$ be a family of graphs. A subgraph $J$ of $G$ is ${\\mathcal F}$-saturated if no member of ${\\mathcal F}$ is a subgraph of $J$, but for any edge $e$ in $E(G)-E(J)$, some element of ${\\mathcal F}$ is a subgraph of $J+e$. We let $\\text{ex}({\\mathcal F},G)$ and $\\text{sat}({\\mathcal F},G)$ denote the maximum and minimum size of an ${\\mathcal F}$-saturated subgraph of $G$, respectively. If no element of ${\\mathcal F}$ is a subgraph of $G$, then $\\text{sat}({\\mathcal F},G) = \\text{ex}({\\mathcal F}, G) = |E(G)|$.\n  In this paper, for $k\\ge 3$ and $n\\ge 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3137","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":"1408.3137","created_at":"2026-05-18T02:45:13.717971+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.3137v1","created_at":"2026-05-18T02:45:13.717971+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3137","created_at":"2026-05-18T02:45:13.717971+00:00"},{"alias_kind":"pith_short_12","alias_value":"AIQSDWK5WXXZ","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_16","alias_value":"AIQSDWK5WXXZZDQD","created_at":"2026-05-18T12:28:19.803747+00:00"},{"alias_kind":"pith_short_8","alias_value":"AIQSDWK5","created_at":"2026-05-18T12:28:19.803747+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/AIQSDWK5WXXZZDQD6QZAFFPKEA","json":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA.json","graph_json":"https://pith.science/api/pith-number/AIQSDWK5WXXZZDQD6QZAFFPKEA/graph.json","events_json":"https://pith.science/api/pith-number/AIQSDWK5WXXZZDQD6QZAFFPKEA/events.json","paper":"https://pith.science/paper/AIQSDWK5"},"agent_actions":{"view_html":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA","download_json":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA.json","view_paper":"https://pith.science/paper/AIQSDWK5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.3137&json=true","fetch_graph":"https://pith.science/api/pith-number/AIQSDWK5WXXZZDQD6QZAFFPKEA/graph.json","fetch_events":"https://pith.science/api/pith-number/AIQSDWK5WXXZZDQD6QZAFFPKEA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA/action/storage_attestation","attest_author":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA/action/author_attestation","sign_citation":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA/action/citation_signature","submit_replication":"https://pith.science/pith/AIQSDWK5WXXZZDQD6QZAFFPKEA/action/replication_record"}},"created_at":"2026-05-18T02:45:13.717971+00:00","updated_at":"2026-05-18T02:45:13.717971+00:00"}