{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:J545AMYMPBSITVVRXQP53UAZ4K","short_pith_number":"pith:J545AMYM","canonical_record":{"source":{"id":"2506.02687","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T09:40:28Z","cross_cats_sorted":[],"title_canon_sha256":"4f9d0afec86ddedbe86756cb1e962bb5c9f3106d09250c8d110cb69863337858","abstract_canon_sha256":"d3b2a8bb4d833f4478528961e726a09f7814f4d193678c8ab5084ea03f259e36"},"schema_version":"1.0"},"canonical_sha256":"4f79d0330c786489d6b1bc1fddd019e2a293ba718e38d33855784e53f3f8f8b8","source":{"kind":"arxiv","id":"2506.02687","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.02687","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"arxiv_version","alias_value":"2506.02687v3","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.02687","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_12","alias_value":"J545AMYMPBSI","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_16","alias_value":"J545AMYMPBSITVVR","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_8","alias_value":"J545AMYM","created_at":"2026-07-05T11:19:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:J545AMYMPBSITVVRXQP53UAZ4K","target":"record","payload":{"canonical_record":{"source":{"id":"2506.02687","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T09:40:28Z","cross_cats_sorted":[],"title_canon_sha256":"4f9d0afec86ddedbe86756cb1e962bb5c9f3106d09250c8d110cb69863337858","abstract_canon_sha256":"d3b2a8bb4d833f4478528961e726a09f7814f4d193678c8ab5084ea03f259e36"},"schema_version":"1.0"},"canonical_sha256":"4f79d0330c786489d6b1bc1fddd019e2a293ba718e38d33855784e53f3f8f8b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:19:27.961523Z","signature_b64":"PrAR4bsVOiFOcpomyRLoKOq7Iy+CT1PTusbpwKTdS8Fl+CnCqgKfqovOmB3INBCL2gkekrEiTuPGZVUmafkyAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f79d0330c786489d6b1bc1fddd019e2a293ba718e38d33855784e53f3f8f8b8","last_reissued_at":"2026-07-05T11:19:27.961040Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:19:27.961040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.02687","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:19:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2sZY0aOuR1MDpQKQv9Duo2Qt2EuQYUtdemv8RmTrqIyUiSwrqWYMgFFPR0d3y5SU6R7Qq4ZfJUK8VH7hRGGyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T02:49:50.506322Z"},"content_sha256":"70738769cd10f1372befe3632863e149d416ff691b7689561297fda33e005843","schema_version":"1.0","event_id":"sha256:70738769cd10f1372befe3632863e149d416ff691b7689561297fda33e005843"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:J545AMYMPBSITVVRXQP53UAZ4K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Fan-type condition involving bipartite independence number for hamiltonicity in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongxi Liu, Long-tu Yuan, Xiaowen Zhang","submitted_at":"2025-06-03T09:40:28Z","abstract_excerpt":"The bipartite independence number of a graph $G$, denoted by $\\widetilde{\\alpha}(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s + t = q + 1$, such that for any two disjoint subsets $A, B \\subseteq V(G)$ with $|A| = s$ and $|B| = t$, there exists an edge between $A$ and $B$. In this paper, we prove that for a 2-connected graph $G$ of order at least three, if $\\max\\{d_G(x), d_G(y)\\} \\ge \\widetilde{\\alpha}(G)$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian. Moreover, we prove that if $G$ is 3-connecte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02687","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.02687/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T11:19:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0qJVQxrqgb43e92lwm/pMVLEoFiyn6b+xxVmJLCtZJBz6nAH+yGZt4ctNTfeC4cTqLIUaX2hI1p4sG10w1+tAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T02:49:50.506862Z"},"content_sha256":"a9dde3579dba154511461a09bcda90bdef3cb583c558282b7a17b506c734a1a8","schema_version":"1.0","event_id":"sha256:a9dde3579dba154511461a09bcda90bdef3cb583c558282b7a17b506c734a1a8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/J545AMYMPBSITVVRXQP53UAZ4K/bundle.json","state_url":"https://pith.science/pith/J545AMYMPBSITVVRXQP53UAZ4K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/J545AMYMPBSITVVRXQP53UAZ4K/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":"2026-08-08T02:49:50Z","links":{"resolver":"https://pith.science/pith/J545AMYMPBSITVVRXQP53UAZ4K","bundle":"https://pith.science/pith/J545AMYMPBSITVVRXQP53UAZ4K/bundle.json","state":"https://pith.science/pith/J545AMYMPBSITVVRXQP53UAZ4K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/J545AMYMPBSITVVRXQP53UAZ4K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:J545AMYMPBSITVVRXQP53UAZ4K","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":"d3b2a8bb4d833f4478528961e726a09f7814f4d193678c8ab5084ea03f259e36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T09:40:28Z","title_canon_sha256":"4f9d0afec86ddedbe86756cb1e962bb5c9f3106d09250c8d110cb69863337858"},"schema_version":"1.0","source":{"id":"2506.02687","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.02687","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"arxiv_version","alias_value":"2506.02687v3","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.02687","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_12","alias_value":"J545AMYMPBSI","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_16","alias_value":"J545AMYMPBSITVVR","created_at":"2026-07-05T11:19:27Z"},{"alias_kind":"pith_short_8","alias_value":"J545AMYM","created_at":"2026-07-05T11:19:27Z"}],"graph_snapshots":[{"event_id":"sha256:a9dde3579dba154511461a09bcda90bdef3cb583c558282b7a17b506c734a1a8","target":"graph","created_at":"2026-07-05T11:19:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.02687/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The bipartite independence number of a graph $G$, denoted by $\\widetilde{\\alpha}(G)$, is defined as the smallest integer $q$ for which there exist positive integers $s$ and $t$ with $s + t = q + 1$, such that for any two disjoint subsets $A, B \\subseteq V(G)$ with $|A| = s$ and $|B| = t$, there exists an edge between $A$ and $B$. In this paper, we prove that for a 2-connected graph $G$ of order at least three, if $\\max\\{d_G(x), d_G(y)\\} \\ge \\widetilde{\\alpha}(G)$ for every pair of nonadjacent vertices $x, y$ at distance two, then $G$ is hamiltonian. Moreover, we prove that if $G$ is 3-connecte","authors_text":"Hongxi Liu, Long-tu Yuan, Xiaowen Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T09:40:28Z","title":"A Fan-type condition involving bipartite independence number for hamiltonicity in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.02687","kind":"arxiv","version":3},"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":"sha256:70738769cd10f1372befe3632863e149d416ff691b7689561297fda33e005843","target":"record","created_at":"2026-07-05T11:19:27Z","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":"d3b2a8bb4d833f4478528961e726a09f7814f4d193678c8ab5084ea03f259e36","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-03T09:40:28Z","title_canon_sha256":"4f9d0afec86ddedbe86756cb1e962bb5c9f3106d09250c8d110cb69863337858"},"schema_version":"1.0","source":{"id":"2506.02687","kind":"arxiv","version":3}},"canonical_sha256":"4f79d0330c786489d6b1bc1fddd019e2a293ba718e38d33855784e53f3f8f8b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4f79d0330c786489d6b1bc1fddd019e2a293ba718e38d33855784e53f3f8f8b8","first_computed_at":"2026-07-05T11:19:27.961040Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:19:27.961040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PrAR4bsVOiFOcpomyRLoKOq7Iy+CT1PTusbpwKTdS8Fl+CnCqgKfqovOmB3INBCL2gkekrEiTuPGZVUmafkyAg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:19:27.961523Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.02687","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70738769cd10f1372befe3632863e149d416ff691b7689561297fda33e005843","sha256:a9dde3579dba154511461a09bcda90bdef3cb583c558282b7a17b506c734a1a8"],"state_sha256":"4f076792496806953c2e7fd862170a171eb198e17dad8f872133c7a927436f03"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p6/2+qqy6WYW8oBOw57NHzIDIXfWd4jEXnhU2lEQikkcTvJLRJrE0Ky1zrpJSXV1cBdnOYSTwoM2gzAlPiJ3Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T02:49:50.510855Z","bundle_sha256":"5a94a84c660b96a6c2c25298a2e4fe19f8bb171db39263a58815d7f02e687549"}}