{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7OPZZHVA3USOGBKMOYIXHCEAHD","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":"1f088674349c253b66360f57fa18a2ece8fc3713ca0fbe76fdeeec18dddf4fc3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T12:11:19Z","title_canon_sha256":"b968d091e0401b3de0b14eef9a7facfaec6e10ae1303a75a3bf89acbff767e30"},"schema_version":"1.0","source":{"id":"2606.22006","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22006","created_at":"2026-06-23T02:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22006v1","created_at":"2026-06-23T02:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22006","created_at":"2026-06-23T02:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"7OPZZHVA3USO","created_at":"2026-06-23T02:13:05Z"},{"alias_kind":"pith_short_16","alias_value":"7OPZZHVA3USOGBKM","created_at":"2026-06-23T02:13:05Z"},{"alias_kind":"pith_short_8","alias_value":"7OPZZHVA","created_at":"2026-06-23T02:13:05Z"}],"graph_snapshots":[{"event_id":"sha256:6d5cda535ea4c013a258e87c9a8b4f1df403a43752c3d9fea9f73f91340d0ed0","target":"graph","created_at":"2026-06-23T02:13:05Z","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/2606.22006/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $H$ be a graph. A graph $G$ is $H$-saturated if $G$ is $H$-free, but adding any edge between two non-adjacent vertices of $G$ yields an $H$-copy as a subgraph. The saturation number $\\mathrm{sat}(n, H)$ is the minimum number of edges in an $H$-saturated graph on $n$ vertices. The saturation number for the join of a vertex and a graph $F$, denoted by $K_1\\vee F$, has attracted considerable attention. Cameron and Puleo \\cite{Ca} proved that $\\mathrm{sat}(n,K_1 \\vee F)\\le n-1+\\mathrm{sat}(n-1, F)$ for $n > |V(F)|$. A natural question is when the above equality holds. Most existing results imp","authors_text":"Xinying Hua, Yuejian Peng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T12:11:19Z","title":"Saturation numbers of some joins of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22006","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":"sha256:06a06fa5c14e8e93647b435a77397bc67cc0cf96f2ba0a8f604fd38d9d760c1e","target":"record","created_at":"2026-06-23T02:13:05Z","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":"1f088674349c253b66360f57fa18a2ece8fc3713ca0fbe76fdeeec18dddf4fc3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-20T12:11:19Z","title_canon_sha256":"b968d091e0401b3de0b14eef9a7facfaec6e10ae1303a75a3bf89acbff767e30"},"schema_version":"1.0","source":{"id":"2606.22006","kind":"arxiv","version":1}},"canonical_sha256":"fb9f9c9ea0dd24e3054c761173888038c1081ad3e41ffec9e4d96807e3837564","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb9f9c9ea0dd24e3054c761173888038c1081ad3e41ffec9e4d96807e3837564","first_computed_at":"2026-06-23T02:13:05.433685Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T02:13:05.433685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pFHjUJ20yaVNfyJxvk2DKWiFGcn8u8w8xEOOuy3Zf2CxA7tIdCauOSYIsSD+abvcL1s3hF+Hc0Cbfv+2HB4gBg==","signature_status":"signed_v1","signed_at":"2026-06-23T02:13:05.434022Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.22006","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06a06fa5c14e8e93647b435a77397bc67cc0cf96f2ba0a8f604fd38d9d760c1e","sha256:6d5cda535ea4c013a258e87c9a8b4f1df403a43752c3d9fea9f73f91340d0ed0"],"state_sha256":"d75b87c1887cdd9b546f9d43326d6892661d51a8a06241f20d5085bfa8937c15"}