{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:L2LG6TP6U2YDMRZ6ELKUXY7AXR","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":"49feec2ac49331c1249aff34c27ec27f10286d4f1090193037e728b72be202af","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:00:40Z","title_canon_sha256":"6c4075a4f128689389eb910a2b16e5d651e7456e55e25521aef192e30035a22b"},"schema_version":"1.0","source":{"id":"2606.09736","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.09736","created_at":"2026-06-09T02:09:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.09736v1","created_at":"2026-06-09T02:09:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.09736","created_at":"2026-06-09T02:09:06Z"},{"alias_kind":"pith_short_12","alias_value":"L2LG6TP6U2YD","created_at":"2026-06-09T02:09:06Z"},{"alias_kind":"pith_short_16","alias_value":"L2LG6TP6U2YDMRZ6","created_at":"2026-06-09T02:09:06Z"},{"alias_kind":"pith_short_8","alias_value":"L2LG6TP6","created_at":"2026-06-09T02:09:06Z"}],"graph_snapshots":[{"event_id":"sha256:0b14e94466fc6d5f8a835d90b3d9f9043b74a790042a114b5b0dee779d558fac","target":"graph","created_at":"2026-06-09T02:09:06Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.09736/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The triangle packing number $\\nu(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2\\nu(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph $F$. In particular, we show the existence of almost-perfect packings for an infinite family of $F$, and state some negative results a","authors_text":"Andrzej Dudek, Patrick Bennett, Ryan Cushman, Xavier P\\'erez-Gim\\'enez","cross_cats":["cs.DM"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:00:40Z","title":"Almost-perfect packings and Tuza's conjecture in the random geometric graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09736","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:42034884e45fe85078c7678ca8a717266e435a1f0a1ef2c6a232097c500d70ae","target":"record","created_at":"2026-06-09T02:09:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"49feec2ac49331c1249aff34c27ec27f10286d4f1090193037e728b72be202af","cross_cats_sorted":["cs.DM"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-08T17:00:40Z","title_canon_sha256":"6c4075a4f128689389eb910a2b16e5d651e7456e55e25521aef192e30035a22b"},"schema_version":"1.0","source":{"id":"2606.09736","kind":"arxiv","version":1}},"canonical_sha256":"5e966f4dfea6b036473e22d54be3e0bc53b136c99fc123e4d71c3ef9a7e0f280","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5e966f4dfea6b036473e22d54be3e0bc53b136c99fc123e4d71c3ef9a7e0f280","first_computed_at":"2026-06-09T02:09:06.820827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T02:09:06.820827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F67rkj6Ac/0J7xL83a9k9jsErAuiOte6FUphMEy3kfAy5F7lQle1swA3kxAqg41ksugIzocYoIpf75DLTBLoCg==","signature_status":"signed_v1","signed_at":"2026-06-09T02:09:06.821748Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.09736","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:42034884e45fe85078c7678ca8a717266e435a1f0a1ef2c6a232097c500d70ae","sha256:0b14e94466fc6d5f8a835d90b3d9f9043b74a790042a114b5b0dee779d558fac"],"state_sha256":"0e304ecb58fe0e094d0781b36e25cd3224d936724604e1cd149fa0525b382c6f"}