{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:3R4IWEPKRY4D6FH6X4JYW4QNBD","short_pith_number":"pith:3R4IWEPK","canonical_record":{"source":{"id":"1601.03471","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T03:01:20Z","cross_cats_sorted":[],"title_canon_sha256":"eb5f24aea80c87f1d6895e8be620d113888d1955b872accf9ac5e52d3bc0cd9b","abstract_canon_sha256":"d322be81af466eaca9c8cdaa5db33d0c34743e5a18adfb6bb0338d7bc167bb64"},"schema_version":"1.0"},"canonical_sha256":"dc788b11ea8e383f14febf138b720d08cbc71f58a94e4a3cc7a391d94ac1e813","source":{"kind":"arxiv","id":"1601.03471","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03471","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03471v2","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03471","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"pith_short_12","alias_value":"3R4IWEPKRY4D","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3R4IWEPKRY4D6FH6","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3R4IWEPK","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:3R4IWEPKRY4D6FH6X4JYW4QNBD","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03471","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T03:01:20Z","cross_cats_sorted":[],"title_canon_sha256":"eb5f24aea80c87f1d6895e8be620d113888d1955b872accf9ac5e52d3bc0cd9b","abstract_canon_sha256":"d322be81af466eaca9c8cdaa5db33d0c34743e5a18adfb6bb0338d7bc167bb64"},"schema_version":"1.0"},"canonical_sha256":"dc788b11ea8e383f14febf138b720d08cbc71f58a94e4a3cc7a391d94ac1e813","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:04.671633Z","signature_b64":"m1dCbsAXPOafBN3Uh0wEtQ9cCPzwVFCJZ0mtjLdTdckKAaGN5Rpr3k4WFRcOMf1am0Nn+ic7q8nCf69zv0bLDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc788b11ea8e383f14febf138b720d08cbc71f58a94e4a3cc7a391d94ac1e813","last_reissued_at":"2026-05-18T00:19:04.671035Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:04.671035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03471","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:19:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zy4yPj/AT36NkCMmd0K7RQ7Q61Gf7oHbnc5kLZrDwOOkcxi+QRLOKRAadcMQtgFoUZ8rFRLDIlqJlGovO6DUAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:39:23.057440Z"},"content_sha256":"8fd85124498d9c50851453ced4035adf5ceef2539a554379e8bf6eb23fd07ffe","schema_version":"1.0","event_id":"sha256:8fd85124498d9c50851453ced4035adf5ceef2539a554379e8bf6eb23fd07ffe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:3R4IWEPKRY4D6FH6X4JYW4QNBD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Total perfect codes in Cayley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sanming Zhou","submitted_at":"2016-01-14T03:01:20Z","abstract_excerpt":"A total perfect code in a graph $\\Gamma$ is a subset $C$ of $V(\\Gamma)$ such that every vertex of $\\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian $2$-group admits a total perfect code if and only if its degree is a power of $2$. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03471","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:19:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AJz9b/0Q2DHg3DQd5TDNdJ+me2W7CFC57nFbrnb9rYWW6THWYmyRM1k0jG3FUYT3DxxoZg65i7WxsFZQv2yvDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:39:23.058034Z"},"content_sha256":"85de597864f41604a82310e9e73bf1e9dd09ac3e71e724305c2ea82d5cc16f9c","schema_version":"1.0","event_id":"sha256:85de597864f41604a82310e9e73bf1e9dd09ac3e71e724305c2ea82d5cc16f9c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/bundle.json","state_url":"https://pith.science/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T23:39:23Z","links":{"resolver":"https://pith.science/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD","bundle":"https://pith.science/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/bundle.json","state":"https://pith.science/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3R4IWEPKRY4D6FH6X4JYW4QNBD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3R4IWEPKRY4D6FH6X4JYW4QNBD","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":"d322be81af466eaca9c8cdaa5db33d0c34743e5a18adfb6bb0338d7bc167bb64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T03:01:20Z","title_canon_sha256":"eb5f24aea80c87f1d6895e8be620d113888d1955b872accf9ac5e52d3bc0cd9b"},"schema_version":"1.0","source":{"id":"1601.03471","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03471","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03471v2","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03471","created_at":"2026-05-18T00:19:04Z"},{"alias_kind":"pith_short_12","alias_value":"3R4IWEPKRY4D","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"3R4IWEPKRY4D6FH6","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"3R4IWEPK","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:85de597864f41604a82310e9e73bf1e9dd09ac3e71e724305c2ea82d5cc16f9c","target":"graph","created_at":"2026-05-18T00:19:04Z","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"},"paper":{"abstract_excerpt":"A total perfect code in a graph $\\Gamma$ is a subset $C$ of $V(\\Gamma)$ such that every vertex of $\\Gamma$ is adjacent to exactly one vertex in $C$. We give necessary and sufficient conditions for a conjugation-closed subset of a group to be a total perfect code in a Cayley graph of the group. As an application we show that a Cayley graph on an elementary abelian $2$-group admits a total perfect code if and only if its degree is a power of $2$. We also obtain necessary conditions for a Cayley graph of a group with connection set closed under conjugation to admit a total perfect code.","authors_text":"Sanming Zhou","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T03:01:20Z","title":"Total perfect codes in Cayley graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03471","kind":"arxiv","version":2},"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":"sha256:8fd85124498d9c50851453ced4035adf5ceef2539a554379e8bf6eb23fd07ffe","target":"record","created_at":"2026-05-18T00:19:04Z","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":"d322be81af466eaca9c8cdaa5db33d0c34743e5a18adfb6bb0338d7bc167bb64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-14T03:01:20Z","title_canon_sha256":"eb5f24aea80c87f1d6895e8be620d113888d1955b872accf9ac5e52d3bc0cd9b"},"schema_version":"1.0","source":{"id":"1601.03471","kind":"arxiv","version":2}},"canonical_sha256":"dc788b11ea8e383f14febf138b720d08cbc71f58a94e4a3cc7a391d94ac1e813","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc788b11ea8e383f14febf138b720d08cbc71f58a94e4a3cc7a391d94ac1e813","first_computed_at":"2026-05-18T00:19:04.671035Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:04.671035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m1dCbsAXPOafBN3Uh0wEtQ9cCPzwVFCJZ0mtjLdTdckKAaGN5Rpr3k4WFRcOMf1am0Nn+ic7q8nCf69zv0bLDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:04.671633Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03471","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8fd85124498d9c50851453ced4035adf5ceef2539a554379e8bf6eb23fd07ffe","sha256:85de597864f41604a82310e9e73bf1e9dd09ac3e71e724305c2ea82d5cc16f9c"],"state_sha256":"e3b91c80a5ab611b6c2667d872af4b3b6446e2f4e9ab783eb033b3df04b9e06d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/JL9JY6qjpFTCAEPr+sgiH8Hd7Il60DYn0tZpTMWOoZmzB1f5bQ+wrM4/IzAGI46hNFEabTY5musdXHs7RKPAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T23:39:23.060183Z","bundle_sha256":"36bd10a2c0f8f99d03ad2171e6f0b3caab0ba6fa3d856168f4558dc1f8a257b7"}}