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It was conjectured by Lehel and independently by Puleo that $\\alpha_{1} (G) + \\tau_{B} (G) \\le n^2/4$ for every $n$-vertex graph $G$. Puleo showed that $\\alpha_{1} (G) + \\tau_{B} (G) \\le 5n^2/16$ for every $n$-vertex graph $G$. In this note, we improve the bound by showing that $\\alpha_{1} (G) + \\tau_{B} (G) \\le 4403n^2/15000$ for every $n$-vertex graph $G$."},"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":"1512.06202","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-19T07:45:53Z","cross_cats_sorted":[],"title_canon_sha256":"0deb2bc6d2df7fb2b2f5400f08718129412483978adf7f8bad72221bc2b96deb","abstract_canon_sha256":"b257218b0318e08b8c791fb1eeaf952e61291dec0fc16442ea1a9c4b9737ce07"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:31.041077Z","signature_b64":"8KBbC0hHhTuxxR4W5hGoRp+rdwi3bpswnFy+eeCDn0BEM3s+A1zW1Mg/40rAV7DzKO+CGVLiPNmHFAr5uFoUCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe356cbf50129fe06d084049460350577ec40b70069205b493de7e8335b455f1","last_reissued_at":"2026-05-18T01:04:31.040351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:31.040351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on Bipartite Subgraphs and Triangle-independent Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Honghai Xu","submitted_at":"2015-12-19T07:45:53Z","abstract_excerpt":"Let $\\alpha_{1} (G)$ denote the maximum size of an edge set that contains at most one edge from each triangle of $G$. 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