{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:LS535TOIOIB3EQGKBXTMYBLF3S","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":"d9e708e1aefe1f4c2d49ea8c0ee805d062822eb612f362e2b081d96fbef66cd1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-12-12T13:41:26Z","title_canon_sha256":"8cfb52cddead22181057e0cffe72f2172773d8aab16a1573aabe4a2f6c40d95b"},"schema_version":"1.0","source":{"id":"2112.06236","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.06236","created_at":"2026-07-05T03:40:11Z"},{"alias_kind":"arxiv_version","alias_value":"2112.06236v1","created_at":"2026-07-05T03:40:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.06236","created_at":"2026-07-05T03:40:11Z"},{"alias_kind":"pith_short_12","alias_value":"LS535TOIOIB3","created_at":"2026-07-05T03:40:11Z"},{"alias_kind":"pith_short_16","alias_value":"LS535TOIOIB3EQGK","created_at":"2026-07-05T03:40:11Z"},{"alias_kind":"pith_short_8","alias_value":"LS535TOI","created_at":"2026-07-05T03:40:11Z"}],"graph_snapshots":[{"event_id":"sha256:c38db4a351a6b9db4a2488db3a714eea5e50503f1164ad54e37e49c5368c2e9b","target":"graph","created_at":"2026-07-05T03:40:11Z","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/2112.06236/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a graph $\\Gamma$, a perfect code in $\\Gamma$ is an independent set $C$ of vertices of $\\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code in $\\Gamma$ is a set $C$ of vertices of $\\Gamma$ such that every vertex of $\\Gamma$ is adjacent to a unique vertex in $C$. To study (total) perfect codes in vertex-transitive graphs, we generalize the concept of subgroup (total) perfect code of a finite group introduced in \\cite{HXZ18} as follows: Given a finite group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called","authors_text":"Junyang Zhang, Yuting Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-12-12T13:41:26Z","title":"Perfect codes in vertex-transitive graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.06236","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:f162b2a864dec4819a4bfd8f7e816001db9c7dcc472e21c4d691e8bf63ac1b98","target":"record","created_at":"2026-07-05T03:40:11Z","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":"d9e708e1aefe1f4c2d49ea8c0ee805d062822eb612f362e2b081d96fbef66cd1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-12-12T13:41:26Z","title_canon_sha256":"8cfb52cddead22181057e0cffe72f2172773d8aab16a1573aabe4a2f6c40d95b"},"schema_version":"1.0","source":{"id":"2112.06236","kind":"arxiv","version":1}},"canonical_sha256":"5cbbbecdc87203b240ca0de6cc0565dcbb46b3cdb4ef00732e71e210f8eaa9e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5cbbbecdc87203b240ca0de6cc0565dcbb46b3cdb4ef00732e71e210f8eaa9e8","first_computed_at":"2026-07-05T03:40:11.818152Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:40:11.818152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KXIR8oDvWnkTkrPuVhVmCl3D4jW/J/+ZX1GdUkiazCBZOstptFo2oTd/AEpNSmxNZw+LVaIaZXtn2dYhSvcADQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:40:11.818615Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.06236","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f162b2a864dec4819a4bfd8f7e816001db9c7dcc472e21c4d691e8bf63ac1b98","sha256:c38db4a351a6b9db4a2488db3a714eea5e50503f1164ad54e37e49c5368c2e9b"],"state_sha256":"69f874b321b67dd361ae2e5ccc5e4b426b659c993336147e10ccc64a096af431"}