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Erd\\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with $\\lfloor n^2/4\\rfloor+1$ edges, one can find at least $(1+o(1))\\frac{n^2}{16}$ $K_4$-saturating edges. We construct a graph with only $\\frac{2n^2}{33}$ $K_4$-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least $(1+o(1))\\frac{2n^2}{33}$ $K_4$-saturating edges in an $n$-vertex $K_4$-free graph with $\\lfloor n^2/4\\rfloo"},"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":"1312.5248","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-12-18T18:00:48Z","cross_cats_sorted":[],"title_canon_sha256":"4cb856883ee05902e32e0186b9c61e528dc5b82f66891255acbcf5a4bbb1813f","abstract_canon_sha256":"ee0a3aa8ef91dd5cdb70609374a8c848389e86294906e950e2480ccb484a28f0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:52.234323Z","signature_b64":"yCkeLwf+etH8t97Z2PwlBwBHZTa9jj/Q56t68lmGF9xXYSL9Mreb0jswPwYJD0xwanNk40xWUyLMw7A6fw5vBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55a1b50684a16ce20447590fb4083b6240aba1fa3784fd6d4140526bb95e9e53","last_reissued_at":"2026-05-18T02:35:52.233873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:52.233873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of $K_4$-saturating edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, J\\'ozsef Balogh","submitted_at":"2013-12-18T18:00:48Z","abstract_excerpt":"Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with $\\lfloor n^2/4\\rfloor+1$ edges, one can find at least $(1+o(1))\\frac{n^2}{16}$ $K_4$-saturating edges. We construct a graph with only $\\frac{2n^2}{33}$ $K_4$-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least $(1+o(1))\\frac{2n^2}{33}$ $K_4$-saturating edges in an $n$-vertex $K_4$-free graph with $\\lfloor n^2/4\\rfloo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5248","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":""},"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":"1312.5248","created_at":"2026-05-18T02:35:52.233948+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.5248v2","created_at":"2026-05-18T02:35:52.233948+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.5248","created_at":"2026-05-18T02:35:52.233948+00:00"},{"alias_kind":"pith_short_12","alias_value":"KWQ3KBUEUFWO","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"KWQ3KBUEUFWOEBCH","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"KWQ3KBUE","created_at":"2026-05-18T12:27:51.066281+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/KWQ3KBUEUFWOEBCHLEH3ICB3MJ","json":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ.json","graph_json":"https://pith.science/api/pith-number/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/graph.json","events_json":"https://pith.science/api/pith-number/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/events.json","paper":"https://pith.science/paper/KWQ3KBUE"},"agent_actions":{"view_html":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ","download_json":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ.json","view_paper":"https://pith.science/paper/KWQ3KBUE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.5248&json=true","fetch_graph":"https://pith.science/api/pith-number/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/graph.json","fetch_events":"https://pith.science/api/pith-number/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/action/storage_attestation","attest_author":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/action/author_attestation","sign_citation":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/action/citation_signature","submit_replication":"https://pith.science/pith/KWQ3KBUEUFWOEBCHLEH3ICB3MJ/action/replication_record"}},"created_at":"2026-05-18T02:35:52.233948+00:00","updated_at":"2026-05-18T02:35:52.233948+00:00"}