{"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. Moreover, this bound on p is best possible for 0<d≤1/2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The random graph G(n,p) is generated independently of the fixed dn-regular graph G_d, and the new triangle-weighting lemma for weighted complete graphs applies directly to the edge weights arising in the perturbed graph without further restrictions.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For any dn-regular graph perturbed by G(n,p) with p > 2d/(1+2d), there is whp a triangle packing covering all but o(n²) edges, and the bound is optimal for 0 < d ≤ 1/2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"79388962555c17a49a2a89893c4d334a5f71f9d1a839f9947bfb53d077700ad9"},"source":{"id":"2604.25250","kind":"arxiv","version":2},"verdict":{"id":"a2c964ee-8bc2-410e-ad71-fb2c631acf39","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T16:00:46.631172Z","strongest_claim":"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. Moreover, this bound on p is best possible for 0<d≤1/2.","one_line_summary":"For any dn-regular graph perturbed by G(n,p) with p > 2d/(1+2d), there is whp a triangle packing covering all but o(n²) edges, and the bound is optimal for 0 < d ≤ 1/2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The random graph G(n,p) is generated independently of the fixed dn-regular graph G_d, and the new triangle-weighting lemma for weighted complete graphs applies directly to the edge weights arising in the perturbed graph without further restrictions.","pith_extraction_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."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25250/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T05:34:34.555281Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:15:24.986961Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c3200446c04ca694fc8eeeeb6833ea5cc36dbe386f11a5b9656de6673202f296"},"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"}