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In the random setting, Frankl and R\\\"odl showed that, with high probability, $G(n,p)$ contains a triangle packing covering all but $o(n^2p)$ edges whenever $p\\ge n^{-1/2+\\varepsilon}$.\n  In this paper, we study near-perfect triangle packings in randomly perturbed graphs. We prove that for every $d>0$ and every $p>2d/(1+2d)$, if $G_d$ is a $dn$-regular graph on $n$ vertices, then with high probability the union $G_d\\cup G(n,p)$ contains a triangle pack"},"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":"2604.25250","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-04-28T05:57:26Z","cross_cats_sorted":[],"title_canon_sha256":"6ca353331535e8c67d4a8319be0768a19dfd627c5e93cc352e656771af13d648","abstract_canon_sha256":"eb897e06870c25989ab382d967175c80b6158a1a39a1e1d679ae58a43f41177b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T01:09:36.482980Z","signature_b64":"JJTy+z/KDQLIaSnrx/BBTmdftP3xn3qgZAHpkaLwdcXNkfhedld40ZGMzfLXfsdBbDyY2wTaOvMPzjqtm7k3BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f43f389f84407918a3ce50536fb39408c403e225b4b891ce4d4c39006534dc7","last_reissued_at":"2026-06-11T01:09:36.481940Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T01:09:36.481940Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Triangle packings in randomly perturbed graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A dn-regular graph unioned with random G(n,p) for p above 2d/(1+2d) admits a triangle packing covering all but o(n²) edges with high probability, and the bound is sharp for d at most 1/2.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, Lanchao Wang, Xinbu Cheng, Zhifei Yan","submitted_at":"2026-04-28T05:57:26Z","abstract_excerpt":"The longstanding Nash-Williams conjecture asserts that every $K_3$-divisible graph $G$ with $\\delta(G)\\ge 3n/4$ admits a triangle decomposition. In the random setting, Frankl and R\\\"odl showed that, with high probability, $G(n,p)$ contains a triangle packing covering all but $o(n^2p)$ edges whenever $p\\ge n^{-1/2+\\varepsilon}$.\n  In this paper, we study near-perfect triangle packings in randomly perturbed graphs. We prove that for every $d>0$ and every $p>2d/(1+2d)$, if $G_d$ is a $dn$-regular graph on $n$ vertices, then with high probability the union $G_d\\cup G(n,p)$ contains a triangle pack"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"for every d>0 and every p>2d/(1+2d), if G_d is a dn-regular graph on n vertices, then with high probability the union G_d ∪ G(n,p) contains a triangle packing covering all but o(n²) edges. 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