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Is it always the case that $\\alpha_1(G) + \\tau_1(G) \\leq n^2/4$? We also consider a variant on this conjecture: if $\\tau_B(G)$ is the smallest size of an edge set whose deletion makes $G$ bipartite, does the stronger inequality $\\alpha_1(G) + \\tau_B(G) \\leq n^2/4$ always hold?\n  By considering"},"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":"1408.5176","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-08-22T00:04:23Z","cross_cats_sorted":[],"title_canon_sha256":"e29302b4ca7e9d0a6f5b0b90576168fc96768db91e080705fdc67198fc70b602","abstract_canon_sha256":"87cd87cddbee9365598c9e9723a579e93a5121c4485f6364d37aaba45b67da54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:28.754605Z","signature_b64":"wF+dCPUJtFbxLArIOaZgt8X3t0EU41SUglX+9T554z+RrY/d/KuSpR0kjjRuM8/LJsLLAbTqzblHYRLFiOFmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f58379d53b74c3e5f26206d254c7bafd38daecd2b5db1f14dbe03fccffb249ec","last_reissued_at":"2026-05-18T02:20:28.754040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:28.754040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extremal Aspects of the Erd\\H{o}s--Gallai--Tuza Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gregory J. Puleo","submitted_at":"2014-08-22T00:04:23Z","abstract_excerpt":"Erd\\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\\alpha_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\\alpha_1(G) + \\tau_1(G) \\leq n^2/4$? We also consider a variant on this conjecture: if $\\tau_B(G)$ is the smallest size of an edge set whose deletion makes $G$ bipartite, does the stronger inequality $\\alpha_1(G) + \\tau_B(G) \\leq n^2/4$ always hold?\n  By considering"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5176","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":"1408.5176","created_at":"2026-05-18T02:20:28.754130+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.5176v2","created_at":"2026-05-18T02:20:28.754130+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.5176","created_at":"2026-05-18T02:20:28.754130+00:00"},{"alias_kind":"pith_short_12","alias_value":"6WBXTVJ3OTB6","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_16","alias_value":"6WBXTVJ3OTB6L4TC","created_at":"2026-05-18T12:28:16.859392+00:00"},{"alias_kind":"pith_short_8","alias_value":"6WBXTVJ3","created_at":"2026-05-18T12:28:16.859392+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/6WBXTVJ3OTB6L4TCA3JFJR527U","json":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U.json","graph_json":"https://pith.science/api/pith-number/6WBXTVJ3OTB6L4TCA3JFJR527U/graph.json","events_json":"https://pith.science/api/pith-number/6WBXTVJ3OTB6L4TCA3JFJR527U/events.json","paper":"https://pith.science/paper/6WBXTVJ3"},"agent_actions":{"view_html":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U","download_json":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U.json","view_paper":"https://pith.science/paper/6WBXTVJ3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.5176&json=true","fetch_graph":"https://pith.science/api/pith-number/6WBXTVJ3OTB6L4TCA3JFJR527U/graph.json","fetch_events":"https://pith.science/api/pith-number/6WBXTVJ3OTB6L4TCA3JFJR527U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U/action/storage_attestation","attest_author":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U/action/author_attestation","sign_citation":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U/action/citation_signature","submit_replication":"https://pith.science/pith/6WBXTVJ3OTB6L4TCA3JFJR527U/action/replication_record"}},"created_at":"2026-05-18T02:20:28.754130+00:00","updated_at":"2026-05-18T02:20:28.754130+00:00"}