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In this paper we prove two results: (1) Every 2-connected claw-$o$-heavy graph $G$ is Hamiltonian if every pair of vertices $u,v$ in a subgraph $H\\cong Z_1$ contained in an induced subgraph $Z_2$ of $G$ with $d_{"},"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":"1209.3899","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-18T10:10:07Z","cross_cats_sorted":[],"title_canon_sha256":"de6e911c3818fd8d8f185d5438dd99ed3ea4e072bbe57deff250068b4639fb90","abstract_canon_sha256":"49abd6054b0233f6e798d2f1d80d17477cd88f9cc4551b8848c0c5678e3b5fdc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:18.799731Z","signature_b64":"2PajGbS5sG/HcpA+n3Hb0j8ZoHXeE53bbKyzs1V5kJZHa3VWZdcVo7O0rxWSy70Rvq+sPTgtQ7xqdDio+FujDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22208c578b27d4f9cec889693be4bd6f5391417c39eda3ea0264013f8b0fa709","last_reissued_at":"2026-05-18T03:37:18.799042Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:18.799042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two sufficient conditions for the existence of Hamilton cycles in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bing Chen, Bo Ning, Shenggui Zhang","submitted_at":"2012-09-18T10:10:07Z","abstract_excerpt":"Let $G$ be a graph on $n\\geq 3$ vertices, claw the bipartite graph $K_{1,3}$, and $Z_i$ the graph obtained from a triangle by attaching a path of length $i$ to its one vertex. $G$ is called 1-heavy if at least one end vertex of each induced claw of $G$ has degree at least $n/2$, and claw-\\emph{o}-heavy if each induced claw of it has a pair of end vertices with degree sum at least $n$. 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