{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XMCWUT4QGCCDODRPBLG3MD6EQ4","short_pith_number":"pith:XMCWUT4Q","canonical_record":{"source":{"id":"1204.4251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-19T05:13:05Z","cross_cats_sorted":[],"title_canon_sha256":"b3349c89e62fa16e4440e3947163717823228f7565f751e7104804d06aa2ad48","abstract_canon_sha256":"5e5294cc0f916ba900c38fff47e06be3edc29131d20e0ec84c9732f450cae7c9"},"schema_version":"1.0"},"canonical_sha256":"bb056a4f903084370e2f0acdb60fc48722861ea22cee709dc5fc33b91673a537","source":{"kind":"arxiv","id":"1204.4251","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4251","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4251v1","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4251","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"pith_short_12","alias_value":"XMCWUT4QGCCD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XMCWUT4QGCCDODRP","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XMCWUT4Q","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XMCWUT4QGCCDODRPBLG3MD6EQ4","target":"record","payload":{"canonical_record":{"source":{"id":"1204.4251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-19T05:13:05Z","cross_cats_sorted":[],"title_canon_sha256":"b3349c89e62fa16e4440e3947163717823228f7565f751e7104804d06aa2ad48","abstract_canon_sha256":"5e5294cc0f916ba900c38fff47e06be3edc29131d20e0ec84c9732f450cae7c9"},"schema_version":"1.0"},"canonical_sha256":"bb056a4f903084370e2f0acdb60fc48722861ea22cee709dc5fc33b91673a537","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:57:28.025964Z","signature_b64":"G9STzgDW9LS0lN0YwMAFRRHv8pUNwS5ozAVrGy1q1UtXEBpiaBZixtwC4p0D0IH80CGcgGl1jdPmiuUS8nyKAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb056a4f903084370e2f0acdb60fc48722861ea22cee709dc5fc33b91673a537","last_reissued_at":"2026-05-18T03:57:28.025392Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:57:28.025392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1204.4251","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":"2026-05-18T03:57:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dDUcHSop70oYUOz/lEpfYkRPLwAwAifGsX9kjd6zruX+OeAV0oT5oWioVHFDS+zwgW84OyoVfRW8GL0d5KIeDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T09:56:28.197211Z"},"content_sha256":"ad8dc478136036b7a9c2dbb9db63ba3c9350fb435768e89f7dfe732630c64dcb","schema_version":"1.0","event_id":"sha256:ad8dc478136036b7a9c2dbb9db63ba3c9350fb435768e89f7dfe732630c64dcb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XMCWUT4QGCCDODRPBLG3MD6EQ4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fault-tolerant analysis of augmented cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun-Ming Xu, Meijie Ma, Yaxing Song","submitted_at":"2012-04-19T05:13:05Z","abstract_excerpt":"The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\\ge 4)$. This paper determines that the 2-extra connectivity of $AQ_n$ is $6n-17$ for $n\\geq 9$ and the 2-extra edge-connectivity is $6n-9$ for $n\\geq 4$. That is, for $n\\geq 9$ (respectively, $n\\geq 4$), at least $6n-17$ vertices (respectively, $6n-9$ edges) of $AQ_n$ have to be removed to get a disconnected graph that contains no isolated vertices and isolated edges. When the augmented cube is used to model th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4251","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":"2026-05-18T03:57:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3ieTVU9NAbSFM4R7+AguEfwa+Z97S20X9ZecIKpwi3yE1qvAiuikGAA0ObvtPsL3cSsC5yMyujlPcZcQRbCKDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T09:56:28.197617Z"},"content_sha256":"e1d1f9a9751b3b512cf93fc8f1846c983e2226572af1143c588d1d9986a97997","schema_version":"1.0","event_id":"sha256:e1d1f9a9751b3b512cf93fc8f1846c983e2226572af1143c588d1d9986a97997"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/bundle.json","state_url":"https://pith.science/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/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":"2026-06-24T09:56:28Z","links":{"resolver":"https://pith.science/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4","bundle":"https://pith.science/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/bundle.json","state":"https://pith.science/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XMCWUT4QGCCDODRPBLG3MD6EQ4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XMCWUT4QGCCDODRPBLG3MD6EQ4","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":"5e5294cc0f916ba900c38fff47e06be3edc29131d20e0ec84c9732f450cae7c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-19T05:13:05Z","title_canon_sha256":"b3349c89e62fa16e4440e3947163717823228f7565f751e7104804d06aa2ad48"},"schema_version":"1.0","source":{"id":"1204.4251","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.4251","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"arxiv_version","alias_value":"1204.4251v1","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.4251","created_at":"2026-05-18T03:57:28Z"},{"alias_kind":"pith_short_12","alias_value":"XMCWUT4QGCCD","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XMCWUT4QGCCDODRP","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XMCWUT4Q","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:e1d1f9a9751b3b512cf93fc8f1846c983e2226572af1143c588d1d9986a97997","target":"graph","created_at":"2026-05-18T03:57:28Z","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":"The augmented cube $AQ_n$, proposed by Choudum and Sunitha [S. A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71-84], is a $(2n-1)$-regular $(2n-1)$-connected graph $(n\\ge 4)$. This paper determines that the 2-extra connectivity of $AQ_n$ is $6n-17$ for $n\\geq 9$ and the 2-extra edge-connectivity is $6n-9$ for $n\\geq 4$. That is, for $n\\geq 9$ (respectively, $n\\geq 4$), at least $6n-17$ vertices (respectively, $6n-9$ edges) of $AQ_n$ have to be removed to get a disconnected graph that contains no isolated vertices and isolated edges. When the augmented cube is used to model th","authors_text":"Jun-Ming Xu, Meijie Ma, Yaxing Song","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-19T05:13:05Z","title":"Fault-tolerant analysis of augmented cubes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4251","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:ad8dc478136036b7a9c2dbb9db63ba3c9350fb435768e89f7dfe732630c64dcb","target":"record","created_at":"2026-05-18T03:57:28Z","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":"5e5294cc0f916ba900c38fff47e06be3edc29131d20e0ec84c9732f450cae7c9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-19T05:13:05Z","title_canon_sha256":"b3349c89e62fa16e4440e3947163717823228f7565f751e7104804d06aa2ad48"},"schema_version":"1.0","source":{"id":"1204.4251","kind":"arxiv","version":1}},"canonical_sha256":"bb056a4f903084370e2f0acdb60fc48722861ea22cee709dc5fc33b91673a537","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb056a4f903084370e2f0acdb60fc48722861ea22cee709dc5fc33b91673a537","first_computed_at":"2026-05-18T03:57:28.025392Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:57:28.025392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G9STzgDW9LS0lN0YwMAFRRHv8pUNwS5ozAVrGy1q1UtXEBpiaBZixtwC4p0D0IH80CGcgGl1jdPmiuUS8nyKAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:57:28.025964Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.4251","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad8dc478136036b7a9c2dbb9db63ba3c9350fb435768e89f7dfe732630c64dcb","sha256:e1d1f9a9751b3b512cf93fc8f1846c983e2226572af1143c588d1d9986a97997"],"state_sha256":"a7d827e83deb5a495cff493fd9301259bc125b0c57b0953dcbe36b10b3623c37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZWUUJGnnDYbVdyA+IcrLtilAIrSLbApLCnaSUmWEC6Ad9KjlCLWI7msDHMQuBKgF+4oB2dp3dbMbsxAHj8PuCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T09:56:28.199550Z","bundle_sha256":"6cd09a8b7b1e8420622bf40d7429a5cc83262339ae239580c6d89ef2a50c6010"}}