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Is it always the case that $\\alpha_1(G) + \\tau_1(G) \\leq n^2/4$? We have two main results. We first obtain the upper bound $\\alpha_1(G) + \\tau_1(G) \\leq 5n^2/16$, as a partial result towards the Erd\\H{o}s--Gallai--Tuza conjecture. We also show that always $\\alpha_1(G) \\leq n^2/2 - m$, where $m"},"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":"1311.5332","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-11-21T08:36:27Z","cross_cats_sorted":[],"title_canon_sha256":"a1f8ea53d748218c7f078451bab7cd91726a0c53ab3caff5ea6c6d1c711d99aa","abstract_canon_sha256":"fdc852daba9c339dea41c4574a79297484f529332f4d9ffe0abfaa8570d3e296"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:23.979554Z","signature_b64":"jJjH/19jDNpwJcxvKXW/vKDTNnaeiIxvvtB1ws/n0NpCnZ09B7L0EOsuMNU9QM8sj5b7ihuwHvMEG4yYOzoEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aba0f7526caca612dfa9425a14171da5348430a76b9c1a3be7afa2fa93ee3f84","last_reissued_at":"2026-05-18T02:39:23.979170Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:23.979170Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a Conjecture of Erd\\H{o}s, Gallai, and Tuza","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gregory J. Puleo","submitted_at":"2013-11-21T08:36:27Z","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 have two main results. We first obtain the upper bound $\\alpha_1(G) + \\tau_1(G) \\leq 5n^2/16$, as a partial result towards the Erd\\H{o}s--Gallai--Tuza conjecture. 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