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They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\\widetilde{\\alpha}(G)\\le a$ and $\\delta(G)\\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=\\Theta\\left(a+k+\\frac{ak}{\\log(k+2)}\\right)."},"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":"2607.05027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T13:07:25Z","cross_cats_sorted":[],"title_canon_sha256":"48ccbf24bad48fc853f399d93c26ad7540d5f63fd7a27fb8ddb5cd4daeabbaa6","abstract_canon_sha256":"d0d146dd739a8237cfc00946970c4a1863dbb5530a5ef112b9c75c238c20256e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:20:19.211099Z","signature_b64":"/xurfIPiXe1gJ8qQPK9vmfFD2FpwrO8OGa/KOtUu4iICTKU2FYZKaX4Oz6hqWz/5f6l4rG4ArENPdFzueyfoAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e781eeff5f4167297ad191eac32f6d75919029b0cfb6e4d695fc3c646340c526","last_reissued_at":"2026-07-07T02:20:19.210324Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:20:19.210324Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Edge-disjoint Hamilton cycles under a bipartite-hole condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chengli Li, Feng Liu, Yanan Hu","submitted_at":"2026-07-06T13:07:25Z","abstract_excerpt":"In 2017, McDiarmid and Yolov introduced the bipartite-hole-number $\\widetilde{\\alpha}(G)$ and proved that $\\delta(G)\\ge \\widetilde{\\alpha}(G)$ forces a Hamilton cycle. They also gave a sufficient condition for packing edge-disjoint Hamilton cycles, and asked whether this condition is sharp or can be relaxed. For integers $a,k\\ge 2$, let $f(a,k)$ be the least integer $d$ such that every graph $G$ on at least three vertices with $\\widetilde{\\alpha}(G)\\le a$ and $\\delta(G)\\ge d$ contains $k$ pairwise edge-disjoint Hamilton cycles. We prove that $f(a,k)=\\Theta\\left(a+k+\\frac{ak}{\\log(k+2)}\\right)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05027","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05027/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2607.05027","created_at":"2026-07-07T02:20:19.210458+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.05027v1","created_at":"2026-07-07T02:20:19.210458+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05027","created_at":"2026-07-07T02:20:19.210458+00:00"},{"alias_kind":"pith_short_12","alias_value":"46A65727IFTS","created_at":"2026-07-07T02:20:19.210458+00:00"},{"alias_kind":"pith_short_16","alias_value":"46A65727IFTSS6WR","created_at":"2026-07-07T02:20:19.210458+00:00"},{"alias_kind":"pith_short_8","alias_value":"46A65727","created_at":"2026-07-07T02:20:19.210458+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/46A65727IFTSS6WRSHVMGL3NOW","json":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW.json","graph_json":"https://pith.science/api/pith-number/46A65727IFTSS6WRSHVMGL3NOW/graph.json","events_json":"https://pith.science/api/pith-number/46A65727IFTSS6WRSHVMGL3NOW/events.json","paper":"https://pith.science/paper/46A65727"},"agent_actions":{"view_html":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW","download_json":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW.json","view_paper":"https://pith.science/paper/46A65727","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.05027&json=true","fetch_graph":"https://pith.science/api/pith-number/46A65727IFTSS6WRSHVMGL3NOW/graph.json","fetch_events":"https://pith.science/api/pith-number/46A65727IFTSS6WRSHVMGL3NOW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW/action/storage_attestation","attest_author":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW/action/author_attestation","sign_citation":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW/action/citation_signature","submit_replication":"https://pith.science/pith/46A65727IFTSS6WRSHVMGL3NOW/action/replication_record"}},"created_at":"2026-07-07T02:20:19.210458+00:00","updated_at":"2026-07-07T02:20:19.210458+00:00"}