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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2112.06236","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-12-12T13:41:26Z","cross_cats_sorted":[],"title_canon_sha256":"8cfb52cddead22181057e0cffe72f2172773d8aab16a1573aabe4a2f6c40d95b","abstract_canon_sha256":"d9e708e1aefe1f4c2d49ea8c0ee805d062822eb612f362e2b081d96fbef66cd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:40:11.818615Z","signature_b64":"KXIR8oDvWnkTkrPuVhVmCl3D4jW/J/+ZX1GdUkiazCBZOstptFo2oTd/AEpNSmxNZw+LVaIaZXtn2dYhSvcADQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5cbbbecdc87203b240ca0de6cc0565dcbb46b3cdb4ef00732e71e210f8eaa9e8","last_reissued_at":"2026-07-05T03:40:11.818152Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:40:11.818152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perfect codes in vertex-transitive graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Junyang Zhang, Yuting Wang","submitted_at":"2021-12-12T13:41:26Z","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 $C$. 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