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Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\\lfloor \\frac{n-1}{2} \\rfloor \\lceil \\frac{n-1}{2} \\rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $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":"1804.10359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","cross_cats_sorted":[],"title_canon_sha256":"ff1c7fedd7e23caa7ac64a660e8807c3a303b92bd09dcb21f8f8036412ad0108","abstract_canon_sha256":"30ae186ac55a701156c397d01d7529b1a0a7e3b38869671d488317054fc570a8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:20.983157Z","signature_b64":"RlKGjEXWw7japcW1FcENYTk9R/F4G5wDG7ED26SMLlGE3N4hXeq7eD3afpCJgega2vh2EgMraAlfdxGe+eTJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","last_reissued_at":"2026-05-18T00:17:20.982450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:20.982450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The edge spectrum of $K_4^-$-saturated graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Gao, Xinmin Hou, Yue Ma","submitted_at":"2018-04-27T06:49:23Z","abstract_excerpt":"Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\\lfloor \\frac{n-1}{2} \\rfloor \\lceil \\frac{n-1}{2} \\rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\\ge 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10359","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":"1804.10359","created_at":"2026-05-18T00:17:20.982541+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.10359v1","created_at":"2026-05-18T00:17:20.982541+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10359","created_at":"2026-05-18T00:17:20.982541+00:00"},{"alias_kind":"pith_short_12","alias_value":"F55J2GEAB2CD","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_16","alias_value":"F55J2GEAB2CDN34I","created_at":"2026-05-18T12:32:22.470017+00:00"},{"alias_kind":"pith_short_8","alias_value":"F55J2GEA","created_at":"2026-05-18T12:32:22.470017+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/F55J2GEAB2CDN34IQZE2OQD4MQ","json":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ.json","graph_json":"https://pith.science/api/pith-number/F55J2GEAB2CDN34IQZE2OQD4MQ/graph.json","events_json":"https://pith.science/api/pith-number/F55J2GEAB2CDN34IQZE2OQD4MQ/events.json","paper":"https://pith.science/paper/F55J2GEA"},"agent_actions":{"view_html":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ","download_json":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ.json","view_paper":"https://pith.science/paper/F55J2GEA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.10359&json=true","fetch_graph":"https://pith.science/api/pith-number/F55J2GEAB2CDN34IQZE2OQD4MQ/graph.json","fetch_events":"https://pith.science/api/pith-number/F55J2GEAB2CDN34IQZE2OQD4MQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/action/storage_attestation","attest_author":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/action/author_attestation","sign_citation":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/action/citation_signature","submit_replication":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/action/replication_record"}},"created_at":"2026-05-18T00:17:20.982541+00:00","updated_at":"2026-05-18T00:17:20.982541+00:00"}