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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1309.5379","created_at":"2026-05-18T03:12:11.136991+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.5379v2","created_at":"2026-05-18T03:12:11.136991+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5379","created_at":"2026-05-18T03:12:11.136991+00:00"},{"alias_kind":"pith_short_12","alias_value":"EPRWC5D4LM2O","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"EPRWC5D4LM2OUSBU","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"EPRWC5D4","created_at":"2026-05-18T12:27:43.054852+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J","json":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J.json","graph_json":"https://pith.science/api/pith-number/EPRWC5D4LM2OUSBUKDGMXYNH6J/graph.json","events_json":"https://pith.science/api/pith-number/EPRWC5D4LM2OUSBUKDGMXYNH6J/events.json","paper":"https://pith.science/paper/EPRWC5D4"},"agent_actions":{"view_html":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J","download_json":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J.json","view_paper":"https://pith.science/paper/EPRWC5D4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.5379&json=true","fetch_graph":"https://pith.science/api/pith-number/EPRWC5D4LM2OUSBUKDGMXYNH6J/graph.json","fetch_events":"https://pith.science/api/pith-number/EPRWC5D4LM2OUSBUKDGMXYNH6J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/action/storage_attestation","attest_author":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/action/author_attestation","sign_citation":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/action/citation_signature","submit_replication":"https://pith.science/pith/EPRWC5D4LM2OUSBUKDGMXYNH6J/action/replication_record"}},"created_at":"2026-05-18T03:12:11.136991+00:00","updated_at":"2026-05-18T03:12:11.136991+00:00"}