{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:WDOHXNNZZ2GKL62FSONACP7IBH","short_pith_number":"pith:WDOHXNNZ","canonical_record":{"source":{"id":"1212.3940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-17T09:17:37Z","cross_cats_sorted":[],"title_canon_sha256":"3ce89b18c70ec8ff2d403e8eb9945b38dd4de1659b20cdacb723292357f345ee","abstract_canon_sha256":"737d18c8351f62596edef8ae71bb63767dde599c42bc87774f113554db45d359"},"schema_version":"1.0"},"canonical_sha256":"b0dc7bb5b9ce8ca5fb45939a013fe809e7bb79a70f9cf5b4dcb2a18a37c74fef","source":{"kind":"arxiv","id":"1212.3940","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3940","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3940v1","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3940","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"pith_short_12","alias_value":"WDOHXNNZZ2GK","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WDOHXNNZZ2GKL62F","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WDOHXNNZ","created_at":"2026-05-18T12:27:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:WDOHXNNZZ2GKL62FSONACP7IBH","target":"record","payload":{"canonical_record":{"source":{"id":"1212.3940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-17T09:17:37Z","cross_cats_sorted":[],"title_canon_sha256":"3ce89b18c70ec8ff2d403e8eb9945b38dd4de1659b20cdacb723292357f345ee","abstract_canon_sha256":"737d18c8351f62596edef8ae71bb63767dde599c42bc87774f113554db45d359"},"schema_version":"1.0"},"canonical_sha256":"b0dc7bb5b9ce8ca5fb45939a013fe809e7bb79a70f9cf5b4dcb2a18a37c74fef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:21.981349Z","signature_b64":"eVD9mzhVAV/y0IOraBU6Lkxpo29YHy9gtxkXsbk6EVGyNrca4lesIj89A04ALCHWrupbxtRrOcSq1FXDm6dWCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0dc7bb5b9ce8ca5fb45939a013fe809e7bb79a70f9cf5b4dcb2a18a37c74fef","last_reissued_at":"2026-05-18T03:38:21.980745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:21.980745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.3940","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:38:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u92v4X4q5zk/d5aGqbuqPLgqTBgOdfDgKJh0vlQxdGNtiBdRpr0iNFbAy1VhnfuTXQcslCYqdKC17FcEh+Z1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:36:56.427404Z"},"content_sha256":"859b4adffda4b2f3143bf9855a8acece3650455a122e0f1c72ede2938a737db4","schema_version":"1.0","event_id":"sha256:859b4adffda4b2f3143bf9855a8acece3650455a122e0f1c72ede2938a737db4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:WDOHXNNZZ2GKL62FSONACP7IBH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"3-Factor-criticality of vertex-transitive graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Heping Zhang, Wuyang Sun","submitted_at":"2012-12-17T09:17:37Z","abstract_excerpt":"A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3940","kind":"arxiv","version":1},"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":"2026-05-18T03:38:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ma4TnvxFzFeDwEDWc4Pj9iN2W3ne0h7Fm0tukZi+FFTRyxvsqj5XeqGkdAmS5IQBf6qygfAGdiTESS5kQJ/6Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T07:36:56.427733Z"},"content_sha256":"85d96b17be322daa6abee41a5a7508c23696974b1e721f36d1e874b3b8d683bd","schema_version":"1.0","event_id":"sha256:85d96b17be322daa6abee41a5a7508c23696974b1e721f36d1e874b3b8d683bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WDOHXNNZZ2GKL62FSONACP7IBH/bundle.json","state_url":"https://pith.science/pith/WDOHXNNZZ2GKL62FSONACP7IBH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WDOHXNNZZ2GKL62FSONACP7IBH/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":"2026-06-20T07:36:56Z","links":{"resolver":"https://pith.science/pith/WDOHXNNZZ2GKL62FSONACP7IBH","bundle":"https://pith.science/pith/WDOHXNNZZ2GKL62FSONACP7IBH/bundle.json","state":"https://pith.science/pith/WDOHXNNZZ2GKL62FSONACP7IBH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WDOHXNNZZ2GKL62FSONACP7IBH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:WDOHXNNZZ2GKL62FSONACP7IBH","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":"737d18c8351f62596edef8ae71bb63767dde599c42bc87774f113554db45d359","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-17T09:17:37Z","title_canon_sha256":"3ce89b18c70ec8ff2d403e8eb9945b38dd4de1659b20cdacb723292357f345ee"},"schema_version":"1.0","source":{"id":"1212.3940","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.3940","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"arxiv_version","alias_value":"1212.3940v1","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.3940","created_at":"2026-05-18T03:38:21Z"},{"alias_kind":"pith_short_12","alias_value":"WDOHXNNZZ2GK","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_16","alias_value":"WDOHXNNZZ2GKL62F","created_at":"2026-05-18T12:27:25Z"},{"alias_kind":"pith_short_8","alias_value":"WDOHXNNZ","created_at":"2026-05-18T12:27:25Z"}],"graph_snapshots":[{"event_id":"sha256:85d96b17be322daa6abee41a5a7508c23696974b1e721f36d1e874b3b8d683bd","target":"graph","created_at":"2026-05-18T03:38:21Z","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 graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if ","authors_text":"Heping Zhang, Wuyang Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-17T09:17:37Z","title":"3-Factor-criticality of vertex-transitive graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3940","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":"sha256:859b4adffda4b2f3143bf9855a8acece3650455a122e0f1c72ede2938a737db4","target":"record","created_at":"2026-05-18T03:38:21Z","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":"737d18c8351f62596edef8ae71bb63767dde599c42bc87774f113554db45d359","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-17T09:17:37Z","title_canon_sha256":"3ce89b18c70ec8ff2d403e8eb9945b38dd4de1659b20cdacb723292357f345ee"},"schema_version":"1.0","source":{"id":"1212.3940","kind":"arxiv","version":1}},"canonical_sha256":"b0dc7bb5b9ce8ca5fb45939a013fe809e7bb79a70f9cf5b4dcb2a18a37c74fef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0dc7bb5b9ce8ca5fb45939a013fe809e7bb79a70f9cf5b4dcb2a18a37c74fef","first_computed_at":"2026-05-18T03:38:21.980745Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:21.980745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eVD9mzhVAV/y0IOraBU6Lkxpo29YHy9gtxkXsbk6EVGyNrca4lesIj89A04ALCHWrupbxtRrOcSq1FXDm6dWCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:21.981349Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.3940","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:859b4adffda4b2f3143bf9855a8acece3650455a122e0f1c72ede2938a737db4","sha256:85d96b17be322daa6abee41a5a7508c23696974b1e721f36d1e874b3b8d683bd"],"state_sha256":"c6e2e85389b2fcd7e89c4a2310c3d80c55379491a143da5b0dc80fb770d9e5b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EUN5SL1A43fGXFBEOVY/KKlWhFJSbUNmSg5e5TXUZrU9FCaI1q5ydVKjUHZZiTO9E3CLnpF1NUvRfvlHfaDODw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T07:36:56.429514Z","bundle_sha256":"297b7630f61746b72a05bb25cda45400ddf50f6b01c6b3b8bef662cb47ddcfe3"}}