{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:AEMIQKLEY5DBHIQ4PFGXF5GEKB","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":"7a089369795d75a914774c0ce973ad6744a8770f45074f079b889f3b52eb37bc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-09T20:02:25Z","title_canon_sha256":"6e0fd7fb386d84064743f0e09f875334d2fdc1c23f8574363b7de8122e667988"},"schema_version":"1.0","source":{"id":"1303.2263","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.2263","created_at":"2026-05-18T01:12:36Z"},{"alias_kind":"arxiv_version","alias_value":"1303.2263v4","created_at":"2026-05-18T01:12:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.2263","created_at":"2026-05-18T01:12:36Z"},{"alias_kind":"pith_short_12","alias_value":"AEMIQKLEY5DB","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_16","alias_value":"AEMIQKLEY5DBHIQ4","created_at":"2026-05-18T12:27:38Z"},{"alias_kind":"pith_short_8","alias_value":"AEMIQKLE","created_at":"2026-05-18T12:27:38Z"}],"graph_snapshots":[{"event_id":"sha256:720396e83d1e6fb95d74b5a3036300b2afa6bbd0e9ff4aa44412c64c56882902","target":"graph","created_at":"2026-05-18T01:12:36Z","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":"In 1984, Fan gave a sufficient condition involving maximum degree of every pair of vertices at distance two for a graph to be Hamiltonian. Motivated by Fan's result, we say that an induced subgraph $H$ of a graph $G$ is $f$-heavy if for every pair of vertices $u,v\\in V(H)$, $d_{H}(u,v)=2$ implies that $\\max\\{d(u),d(v)\\}\\geq n/2$. For a given graph $R$, $G$ is called $R$-$f$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f$-heavy. For a family $\\mathcal{R}$ of graphs, $G$ is $\\mathcal{R}$-$f$-\\emph{heavy} if $G$ is $R$-$f$-heavy for every $R\\in \\mathcal{R}$. In this note we show t","authors_text":"Bo Ning","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-09T20:02:25Z","title":"Fan-type degree condition restricted to triples of induced subgraphs ensuring Hamiltonicity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.2263","kind":"arxiv","version":4},"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:fb00437bb913ab411e6480775caceaccdb3879281d14e723003494e7467b98df","target":"record","created_at":"2026-05-18T01:12:36Z","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":"7a089369795d75a914774c0ce973ad6744a8770f45074f079b889f3b52eb37bc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-09T20:02:25Z","title_canon_sha256":"6e0fd7fb386d84064743f0e09f875334d2fdc1c23f8574363b7de8122e667988"},"schema_version":"1.0","source":{"id":"1303.2263","kind":"arxiv","version":4}},"canonical_sha256":"0118882964c74613a21c794d72f4c450465066ba27468ec8b05b93be074e6e3e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0118882964c74613a21c794d72f4c450465066ba27468ec8b05b93be074e6e3e","first_computed_at":"2026-05-18T01:12:36.746671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:36.746671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9IAWlSJI4C/DO2mCvPIUbnAtPOLSBuzfoeaG1AqqiDOOtvtTLkZMmNCMl+qoRCTU1tmHQ5LufdxbyEHoIy+0Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:36.747243Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.2263","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb00437bb913ab411e6480775caceaccdb3879281d14e723003494e7467b98df","sha256:720396e83d1e6fb95d74b5a3036300b2afa6bbd0e9ff4aa44412c64c56882902"],"state_sha256":"b2c0dc9d7a9036331ed97bf3230a550b8b25b6d13348bd043412a9575cdb2fc1"}