{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EPRWC5D4LM2OUSBUKDGMXYNH6J","short_pith_number":"pith:EPRWC5D4","canonical_record":{"source":{"id":"1309.5379","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-20T20:58:37Z","cross_cats_sorted":[],"title_canon_sha256":"71f3087830164a6d751214d7b48f5ab2e24c0d604540aff29a5e2ac8af6f2c06","abstract_canon_sha256":"95a06bd29de82e6f2c1503326a548cde94d0c9e18987d616d7b6006b77f93d17"},"schema_version":"1.0"},"canonical_sha256":"23e361747c5b34ea483450cccbe1a7f2664f6937fa905aa1b948b02731733eb7","source":{"kind":"arxiv","id":"1309.5379","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5379","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5379v2","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5379","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"EPRWC5D4LM2O","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EPRWC5D4LM2OUSBU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EPRWC5D4","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EPRWC5D4LM2OUSBUKDGMXYNH6J","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5379","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-20T20:58:37Z","cross_cats_sorted":[],"title_canon_sha256":"71f3087830164a6d751214d7b48f5ab2e24c0d604540aff29a5e2ac8af6f2c06","abstract_canon_sha256":"95a06bd29de82e6f2c1503326a548cde94d0c9e18987d616d7b6006b77f93d17"},"schema_version":"1.0"},"canonical_sha256":"23e361747c5b34ea483450cccbe1a7f2664f6937fa905aa1b948b02731733eb7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:11.137690Z","signature_b64":"dRe87SE5zV68nGCsGNAoeKAUpepw6/14NwN5bajEaZkrZNHNN+B9S4ZNBJ2jewUrwGSam+vXFU1xs/FimE2tBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"23e361747c5b34ea483450cccbe1a7f2664f6937fa905aa1b948b02731733eb7","last_reissued_at":"2026-05-18T03:12:11.136853Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:11.136853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5379","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":"2026-05-18T03:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sDfoKYA4kYySIG7fD5sQP/T/RKx5RShAXSdKgdSW6o9FH+ZJGuVdqa/pqz6rvliylJhC5A8aI02zMeo9ok18DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:00:46.938009Z"},"content_sha256":"d3be167ea39f0a474f73c1c299b121121b046b21f4cd40793361e8ab62e8d227","schema_version":"1.0","event_id":"sha256:d3be167ea39f0a474f73c1c299b121121b046b21f4cd40793361e8ab62e8d227"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EPRWC5D4LM2OUSBUKDGMXYNH6J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of a conjecture of Bauer, Fan and Veldman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tri Lai","submitted_at":"2013-09-20T20:58:37Z","abstract_excerpt":"For a 1-tough graph $G$ we define $\\sigma_3(G) = \\min\\{\\deg(u) + \\deg(v)+ \\deg(w):$ $\\{u, v, w\\}$ is an independent set of vertices$\\}$ and $NC2(G)=\\min \\{|N(u)\\cup N(v)|: d(u,v)=2\\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\\geq \\min\\{n,2NC2(G)\\}$ for any 1-tough graph $G$ with $\\sigma_3(G)\\geq n\\geq 3$, where $c(G)$ is the circumference of $G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\\geq\\min\\{n,2NC2(G)+4\\}$.In this paper, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5379","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":"2026-05-18T03:12:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B8C4bWHuoBEtpFHKwmCa03MlyqCx5QvLuXGjwSt2UVZK4xVOUguLDH4y65NAmt8uCZrxFbqA0MsjZDzLyS8MBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T00:00:46.938354Z"},"content_sha256":"0e9dd1e4dfaad6f7b5e7c7d237c7ce08949e0cf7727dc68db01862b382492253","schema_version":"1.0","event_id":"sha256:0e9dd1e4dfaad6f7b5e7c7d237c7ce08949e0cf7727dc68db01862b382492253"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/bundle.json","state_url":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/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":"2026-06-23T00:00:46Z","links":{"resolver":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J","bundle":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/bundle.json","state":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EPRWC5D4LM2OUSBUKDGMXYNH6J","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":"95a06bd29de82e6f2c1503326a548cde94d0c9e18987d616d7b6006b77f93d17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-20T20:58:37Z","title_canon_sha256":"71f3087830164a6d751214d7b48f5ab2e24c0d604540aff29a5e2ac8af6f2c06"},"schema_version":"1.0","source":{"id":"1309.5379","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5379","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5379v2","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5379","created_at":"2026-05-18T03:12:11Z"},{"alias_kind":"pith_short_12","alias_value":"EPRWC5D4LM2O","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EPRWC5D4LM2OUSBU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EPRWC5D4","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:0e9dd1e4dfaad6f7b5e7c7d237c7ce08949e0cf7727dc68db01862b382492253","target":"graph","created_at":"2026-05-18T03:12:11Z","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":"For a 1-tough graph $G$ we define $\\sigma_3(G) = \\min\\{\\deg(u) + \\deg(v)+ \\deg(w):$ $\\{u, v, w\\}$ is an independent set of vertices$\\}$ and $NC2(G)=\\min \\{|N(u)\\cup N(v)|: d(u,v)=2\\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\\geq \\min\\{n,2NC2(G)\\}$ for any 1-tough graph $G$ with $\\sigma_3(G)\\geq n\\geq 3$, where $c(G)$ is the circumference of $G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\\geq\\min\\{n,2NC2(G)+4\\}$.In this paper, we","authors_text":"Tri Lai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-20T20:58:37Z","title":"Proof of a conjecture of Bauer, Fan and Veldman"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5379","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:d3be167ea39f0a474f73c1c299b121121b046b21f4cd40793361e8ab62e8d227","target":"record","created_at":"2026-05-18T03:12:11Z","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":"95a06bd29de82e6f2c1503326a548cde94d0c9e18987d616d7b6006b77f93d17","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-09-20T20:58:37Z","title_canon_sha256":"71f3087830164a6d751214d7b48f5ab2e24c0d604540aff29a5e2ac8af6f2c06"},"schema_version":"1.0","source":{"id":"1309.5379","kind":"arxiv","version":2}},"canonical_sha256":"23e361747c5b34ea483450cccbe1a7f2664f6937fa905aa1b948b02731733eb7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23e361747c5b34ea483450cccbe1a7f2664f6937fa905aa1b948b02731733eb7","first_computed_at":"2026-05-18T03:12:11.136853Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:11.136853Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dRe87SE5zV68nGCsGNAoeKAUpepw6/14NwN5bajEaZkrZNHNN+B9S4ZNBJ2jewUrwGSam+vXFU1xs/FimE2tBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:11.137690Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5379","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3be167ea39f0a474f73c1c299b121121b046b21f4cd40793361e8ab62e8d227","sha256:0e9dd1e4dfaad6f7b5e7c7d237c7ce08949e0cf7727dc68db01862b382492253"],"state_sha256":"6f5888dbc8d4d8e93b122975d83a7c6953553b728550d1801d58f86535b47fd1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KecckY9+qeGC1HeN5inu5P+KppfaWgbiiw5FLYziAxr8Oa0PNFhWOzUVBkZibeXho787OHRBTIUXsGn5iEqmDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T00:00:46.940155Z","bundle_sha256":"1b829481e8b36fcde32af8cc64bb07fc77827173054beadea655093316b460dc"}}