{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:T5B7HCPYIQDZDCR44UCTN6ZZIC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"eb897e06870c25989ab382d967175c80b6158a1a39a1e1d679ae58a43f41177b","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-04-28T05:57:26Z","title_canon_sha256":"6ca353331535e8c67d4a8319be0768a19dfd627c5e93cc352e656771af13d648"},"schema_version":"1.0","source":{"id":"2604.25250","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.25250","created_at":"2026-06-11T01:09:36Z"},{"alias_kind":"arxiv_version","alias_value":"2604.25250v2","created_at":"2026-06-11T01:09:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.25250","created_at":"2026-06-11T01:09:36Z"},{"alias_kind":"pith_short_12","alias_value":"T5B7HCPYIQDZ","created_at":"2026-06-11T01:09:36Z"},{"alias_kind":"pith_short_16","alias_value":"T5B7HCPYIQDZDCR4","created_at":"2026-06-11T01:09:36Z"},{"alias_kind":"pith_short_8","alias_value":"T5B7HCPY","created_at":"2026-06-11T01:09:36Z"}],"graph_snapshots":[{"event_id":"sha256:2a018ed2459e5f0289b9f0afe7a8b8bcbc9adb80402f08a91363c1896a2aff26","target":"graph","created_at":"2026-06-11T01:09:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","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 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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","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."}],"snapshot_sha256":"79388962555c17a49a2a89893c4d334a5f71f9d1a839f9947bfb53d077700ad9"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"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"}],"endpoint":"/pith/2604.25250/integrity.json","findings":[],"snapshot_sha256":"c3200446c04ca694fc8eeeeb6833ea5cc36dbe386f11a5b9656de6673202f296","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Hong Liu, Lanchao Wang, Xinbu Cheng, Zhifei Yan","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-04-28T05:57:26Z","title":"Triangle packings in randomly perturbed graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25250","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T16:00:46.631172Z","id":"a2c964ee-8bc2-410e-ad71-fb2c631acf39","model_set":{"reader":"grok-4.3"},"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","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.","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