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A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has (have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is claw-heavy. In this paper we prove the following two results: (1) Every 2-c"},"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":"1306.5316","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-22T12:52:33Z","cross_cats_sorted":[],"title_canon_sha256":"9986479b82653522e0cafb9777dde90884ec3a2b5913c1e94fed76be6882f2e7","abstract_canon_sha256":"b7977e1a49d02184a0076e62139d61ced631ea93d9075063eb7c59cc7dcdea7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:38.281749Z","signature_b64":"WrY+VSmaD9oddPJMfXq+xn1veBqSb3jO9FBJ26Lr48N5SDyx63KDM+uI4q9GkCMyrq58ZkKeEWiHH+3UGfKODg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f63352c6d0465e885bafa75529261cc8488518a7890f494f208d8d895b037250","last_reissued_at":"2026-05-18T01:12:38.281400Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:38.281400Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hamilton cycles in almost distance-hereditary graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bing Chen, Bo Ning","submitted_at":"2013-06-22T12:52:33Z","abstract_excerpt":"Let $G$ be a graph on $n\\geq 3$ vertices. A graph $G$ is almost distance-hereditary if each connected induced subgraph $H$ of $G$ has the property $d_{H}(x,y)\\leq d_{G}(x,y)+1$ for any pair of vertices $x,y\\in V(H)$. A graph $G$ is called 1-heavy (2-heavy) if at least one (two) of the end vertices of each induced subgraph of $G$ isomorphic to $K_{1,3}$ (a claw) has (have) degree at least $n/2$, and called claw-heavy if each claw of $G$ has a pair of end vertices with degree sum at least $n$. Thus every 2-heavy graph is claw-heavy. In this paper we prove the following two results: (1) Every 2-c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5316","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":"1306.5316","created_at":"2026-05-18T01:12:38.281457+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.5316v2","created_at":"2026-05-18T01:12:38.281457+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5316","created_at":"2026-05-18T01:12:38.281457+00:00"},{"alias_kind":"pith_short_12","alias_value":"6YZVFRWQIZPI","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6YZVFRWQIZPIQW5P","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6YZVFRWQ","created_at":"2026-05-18T12:27:36.564083+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/6YZVFRWQIZPIQW5PU5KSSJQ4ZB","json":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB.json","graph_json":"https://pith.science/api/pith-number/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/graph.json","events_json":"https://pith.science/api/pith-number/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/events.json","paper":"https://pith.science/paper/6YZVFRWQ"},"agent_actions":{"view_html":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB","download_json":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB.json","view_paper":"https://pith.science/paper/6YZVFRWQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.5316&json=true","fetch_graph":"https://pith.science/api/pith-number/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/graph.json","fetch_events":"https://pith.science/api/pith-number/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/action/storage_attestation","attest_author":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/action/author_attestation","sign_citation":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/action/citation_signature","submit_replication":"https://pith.science/pith/6YZVFRWQIZPIQW5PU5KSSJQ4ZB/action/replication_record"}},"created_at":"2026-05-18T01:12:38.281457+00:00","updated_at":"2026-05-18T01:12:38.281457+00:00"}