{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:R7U4NYMYMFWZSRRJ2RUQ34T522","short_pith_number":"pith:R7U4NYMY","canonical_record":{"source":{"id":"1802.09250","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-26T11:29:49Z","cross_cats_sorted":[],"title_canon_sha256":"925c0034e0ca5ad4d085066dff6eac3ca07fb9d90f235d4f8353d829055c3184","abstract_canon_sha256":"f039463c6aca0018701884ac74d19009314aecf1a27d7c6268cc6515c9eab86e"},"schema_version":"1.0"},"canonical_sha256":"8fe9c6e198616d994629d4690df27dd6a3f9b54c512e4d71520f60fe26856a96","source":{"kind":"arxiv","id":"1802.09250","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09250","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09250v1","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09250","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"pith_short_12","alias_value":"R7U4NYMYMFWZ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"R7U4NYMYMFWZSRRJ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"R7U4NYMY","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:R7U4NYMYMFWZSRRJ2RUQ34T522","target":"record","payload":{"canonical_record":{"source":{"id":"1802.09250","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-26T11:29:49Z","cross_cats_sorted":[],"title_canon_sha256":"925c0034e0ca5ad4d085066dff6eac3ca07fb9d90f235d4f8353d829055c3184","abstract_canon_sha256":"f039463c6aca0018701884ac74d19009314aecf1a27d7c6268cc6515c9eab86e"},"schema_version":"1.0"},"canonical_sha256":"8fe9c6e198616d994629d4690df27dd6a3f9b54c512e4d71520f60fe26856a96","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:34.281679Z","signature_b64":"vqA9Pun/fuTddDNETFIBaEUuqF2N7Mt/OXfP306oXAYKrsu7mXcUgIHMbCAM4n0boCP8XTobn7nTTQfXAq/gBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8fe9c6e198616d994629d4690df27dd6a3f9b54c512e4d71520f60fe26856a96","last_reissued_at":"2026-05-18T00:22:34.280972Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:34.280972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1802.09250","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":"2026-05-18T00:22:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"94r4mfGPXKffh8WAfLGWAf8++H4w/yi15XhXAYEVACB9QJB2fW1L8ibXh5/AZAhIkp6sBtQoaVW1eNoa6ltODg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:05:50.353685Z"},"content_sha256":"c3a605a2d17a4ba625ca334bb4e23184d1eb95dbd67f3d110ce037c70b38abf5","schema_version":"1.0","event_id":"sha256:c3a605a2d17a4ba625ca334bb4e23184d1eb95dbd67f3d110ce037c70b38abf5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:R7U4NYMYMFWZSRRJ2RUQ34T522","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The minimum number of Hamilton cycles in a hamiltonian threshold graph of a prescribed order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pu Qiao, Xingzhi Zhan","submitted_at":"2018-02-26T11:29:49Z","abstract_excerpt":"We prove that the minimum number of Hamilton cycles in a hamiltonian threshold graph of order $n$ is $2^{\\lfloor (n-3)/2\\rfloor}$ and this minimum number is attained uniquely by the graph with degree sequence $n-1,n-1,n-2,\\ldots,\\lceil n/2\\rceil,\\lceil n/2\\rceil,\\ldots,3,2$ of $n-2$ distinct degrees. This graph is also the unique graph of minimum size among all hamiltonian threshold graphs of order $n.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09250","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":"2026-05-18T00:22:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I34HLuUedpEA7enjt+odNJkFdCAjjvcMdIuhYhiozA8RIA9kMSo42gcK00CLgxiUa4H0foBxgmwarJ58BL9OCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T01:05:50.354032Z"},"content_sha256":"c4719e69f49aa7d30303bc591eb4e3c747252fce45f040d261d5a5d3cc34cc4c","schema_version":"1.0","event_id":"sha256:c4719e69f49aa7d30303bc591eb4e3c747252fce45f040d261d5a5d3cc34cc4c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/bundle.json","state_url":"https://pith.science/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/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":"2026-06-26T01:05:50Z","links":{"resolver":"https://pith.science/pith/R7U4NYMYMFWZSRRJ2RUQ34T522","bundle":"https://pith.science/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/bundle.json","state":"https://pith.science/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R7U4NYMYMFWZSRRJ2RUQ34T522/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:R7U4NYMYMFWZSRRJ2RUQ34T522","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":"f039463c6aca0018701884ac74d19009314aecf1a27d7c6268cc6515c9eab86e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-26T11:29:49Z","title_canon_sha256":"925c0034e0ca5ad4d085066dff6eac3ca07fb9d90f235d4f8353d829055c3184"},"schema_version":"1.0","source":{"id":"1802.09250","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1802.09250","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"arxiv_version","alias_value":"1802.09250v1","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.09250","created_at":"2026-05-18T00:22:34Z"},{"alias_kind":"pith_short_12","alias_value":"R7U4NYMYMFWZ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"R7U4NYMYMFWZSRRJ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"R7U4NYMY","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:c4719e69f49aa7d30303bc591eb4e3c747252fce45f040d261d5a5d3cc34cc4c","target":"graph","created_at":"2026-05-18T00:22:34Z","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":"We prove that the minimum number of Hamilton cycles in a hamiltonian threshold graph of order $n$ is $2^{\\lfloor (n-3)/2\\rfloor}$ and this minimum number is attained uniquely by the graph with degree sequence $n-1,n-1,n-2,\\ldots,\\lceil n/2\\rceil,\\lceil n/2\\rceil,\\ldots,3,2$ of $n-2$ distinct degrees. This graph is also the unique graph of minimum size among all hamiltonian threshold graphs of order $n.$","authors_text":"Pu Qiao, Xingzhi Zhan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-26T11:29:49Z","title":"The minimum number of Hamilton cycles in a hamiltonian threshold graph of a prescribed order"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.09250","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:c3a605a2d17a4ba625ca334bb4e23184d1eb95dbd67f3d110ce037c70b38abf5","target":"record","created_at":"2026-05-18T00:22:34Z","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":"f039463c6aca0018701884ac74d19009314aecf1a27d7c6268cc6515c9eab86e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-26T11:29:49Z","title_canon_sha256":"925c0034e0ca5ad4d085066dff6eac3ca07fb9d90f235d4f8353d829055c3184"},"schema_version":"1.0","source":{"id":"1802.09250","kind":"arxiv","version":1}},"canonical_sha256":"8fe9c6e198616d994629d4690df27dd6a3f9b54c512e4d71520f60fe26856a96","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8fe9c6e198616d994629d4690df27dd6a3f9b54c512e4d71520f60fe26856a96","first_computed_at":"2026-05-18T00:22:34.280972Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:34.280972Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vqA9Pun/fuTddDNETFIBaEUuqF2N7Mt/OXfP306oXAYKrsu7mXcUgIHMbCAM4n0boCP8XTobn7nTTQfXAq/gBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:34.281679Z","signed_message":"canonical_sha256_bytes"},"source_id":"1802.09250","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3a605a2d17a4ba625ca334bb4e23184d1eb95dbd67f3d110ce037c70b38abf5","sha256:c4719e69f49aa7d30303bc591eb4e3c747252fce45f040d261d5a5d3cc34cc4c"],"state_sha256":"d6351785821ac1feac96d0eee01d956c18aac7a96ba3e4198add40b12186202e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SBIlnmqoszpb59wkdUIxV+LKF0YObj2+hbhKwyDG1cHAqguVc2lUD1aVeJkZFvSy4VaecJLzoH17pQLQkKXuAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T01:05:50.355966Z","bundle_sha256":"b49b4be0b46ee8896682d694f58ac557000c4a2f6b1fe48da1a5ccfdebc86d41"}}