{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:IQR6H5K6M5KDBAXYGCNCKNHMQL","short_pith_number":"pith:IQR6H5K6","schema_version":"1.0","canonical_sha256":"4423e3f55e67543082f8309a2534ec82d6f68868e91f19b945a78783deafd439","source":{"kind":"arxiv","id":"2606.03696","version":1},"attestation_state":"computed","paper":{"title":"Longest cycles and Dirac-type results in highly connected graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Jie Ma, Ziyuan Zhao","submitted_at":"2026-06-02T14:17:06Z","abstract_excerpt":"A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\\le 3$ and remains open for all $k\\ge 4$.\n  In this paper, we prove Bondy's conjecture for all"},"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":"2606.03696","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-02T14:17:06Z","cross_cats_sorted":[],"title_canon_sha256":"365323c232d1da16589f74f09c6b428a55705ebf51a79be8167e9b3c700183e4","abstract_canon_sha256":"93aae30e2c86165c2507c4a55b4661aa8b5d93660d8633a471ea4ec4b8d82a91"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:06:04.778594Z","signature_b64":"u7a7jNavL8frAyAaTdy4SYVgLds4+DxbG0lErGZNMnhQ3IcpbORchE5CuJrznjna3J80vy1KikBCHsVj9KgpCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4423e3f55e67543082f8309a2534ec82d6f68868e91f19b945a78783deafd439","last_reissued_at":"2026-06-03T01:06:04.778170Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:06:04.778170Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Longest cycles and Dirac-type results in highly connected graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Jie Ma, Ziyuan Zhao","submitted_at":"2026-06-02T14:17:06Z","abstract_excerpt":"A classical theorem of Nash-Williams states that if $G$ is a $2$-connected graph on $n$ vertices with minimum degree at least $(n+2)/3$, then for every longest cycle $C$ of $G$, the graph $G-V(C)$ is edgeless. Motivated by a higher-connectivity analogue, Bondy conjectured in 1980 that if $G$ is a $k$-connected graph on $n$ vertices with minimum degree at least $(n+k(k-1))/(k+1)$, then for every longest cycle $C$ of $G$, every path in $G-V(C)$ has at most $k-1$ vertices. This conjecture is known for $k\\le 3$ and remains open for all $k\\ge 4$.\n  In this paper, we prove Bondy's conjecture for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.03696","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/2606.03696/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":"2606.03696","created_at":"2026-06-03T01:06:04.778229+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.03696v1","created_at":"2026-06-03T01:06:04.778229+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.03696","created_at":"2026-06-03T01:06:04.778229+00:00"},{"alias_kind":"pith_short_12","alias_value":"IQR6H5K6M5KD","created_at":"2026-06-03T01:06:04.778229+00:00"},{"alias_kind":"pith_short_16","alias_value":"IQR6H5K6M5KDBAXY","created_at":"2026-06-03T01:06:04.778229+00:00"},{"alias_kind":"pith_short_8","alias_value":"IQR6H5K6","created_at":"2026-06-03T01:06:04.778229+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/IQR6H5K6M5KDBAXYGCNCKNHMQL","json":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL.json","graph_json":"https://pith.science/api/pith-number/IQR6H5K6M5KDBAXYGCNCKNHMQL/graph.json","events_json":"https://pith.science/api/pith-number/IQR6H5K6M5KDBAXYGCNCKNHMQL/events.json","paper":"https://pith.science/paper/IQR6H5K6"},"agent_actions":{"view_html":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL","download_json":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL.json","view_paper":"https://pith.science/paper/IQR6H5K6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.03696&json=true","fetch_graph":"https://pith.science/api/pith-number/IQR6H5K6M5KDBAXYGCNCKNHMQL/graph.json","fetch_events":"https://pith.science/api/pith-number/IQR6H5K6M5KDBAXYGCNCKNHMQL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL/action/storage_attestation","attest_author":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL/action/author_attestation","sign_citation":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL/action/citation_signature","submit_replication":"https://pith.science/pith/IQR6H5K6M5KDBAXYGCNCKNHMQL/action/replication_record"}},"created_at":"2026-06-03T01:06:04.778229+00:00","updated_at":"2026-06-03T01:06:04.778229+00:00"}