{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:V25NXU2UANCYI2QTD5PWA7E4YS","short_pith_number":"pith:V25NXU2U","schema_version":"1.0","canonical_sha256":"aebadbd3540345846a131f5f607c9cc4b35c647292a9416392cd0d40e5bc83fc","source":{"kind":"arxiv","id":"2606.24426","version":1},"attestation_state":"computed","paper":{"title":"Odd cycles in symmetric Cayley graphs on prime cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Yang, Wei Li","submitted_at":"2026-06-23T11:04:15Z","abstract_excerpt":"Let $p$ be an odd prime and let $S\\subseteq \\Z_p$ be symmetric with $0\\notin S$. Let $\\Cay(\\Z_p,S)$ be the undirected Cayley graph on $\\Z_p$ in which $x$ and $y$ are adjacent if and only if $x-y\\in S$. For $1\\le \\ell\\le (p-1)/2$, define \\[ \\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=\\max\\{|S|: S=-S,\\ 0\\notin S,\\ \\Cay(\\Z_p,S)\\text{ contains no }C_{2\\ell+1}\\}. \\] Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\\ell+1$, then $\\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=0$, while if $p>2\\ell+1$, then \\[ \\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=2\\floor{\\frac{p+2\\ell+1}{2(2\\ell+1)}}. \\] The proof combines a sharp additiv"},"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":"2606.24426","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-23T11:04:15Z","cross_cats_sorted":[],"title_canon_sha256":"849c772a8e1eb7adead9c3b23fd2c03729b6fe6e8ea0fb782dbdfd81680715fe","abstract_canon_sha256":"fe9fcca4e07d243ee648daffdeeb9d0553177fe8bfeb6f0b3c6b4d015d8bc17c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-24T01:15:30.102987Z","signature_b64":"omZ1f+C3ncg1ghGCBIv0kzXnvv+Y8AczzPXe3gM7+dClYJJarpM0mBnprxLJkUCougy2UNBygngoUC9e8ADxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aebadbd3540345846a131f5f607c9cc4b35c647292a9416392cd0d40e5bc83fc","last_reissued_at":"2026-06-24T01:15:30.102634Z","signature_status":"signed_v1","first_computed_at":"2026-06-24T01:15:30.102634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Odd cycles in symmetric Cayley graphs on prime cyclic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Yang, Wei Li","submitted_at":"2026-06-23T11:04:15Z","abstract_excerpt":"Let $p$ be an odd prime and let $S\\subseteq \\Z_p$ be symmetric with $0\\notin S$. Let $\\Cay(\\Z_p,S)$ be the undirected Cayley graph on $\\Z_p$ in which $x$ and $y$ are adjacent if and only if $x-y\\in S$. For $1\\le \\ell\\le (p-1)/2$, define \\[ \\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=\\max\\{|S|: S=-S,\\ 0\\notin S,\\ \\Cay(\\Z_p,S)\\text{ contains no }C_{2\\ell+1}\\}. \\] Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\\ell+1$, then $\\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=0$, while if $p>2\\ell+1$, then \\[ \\ex_{\\Cay}(C_{2\\ell+1},\\Z_p)=2\\floor{\\frac{p+2\\ell+1}{2(2\\ell+1)}}. \\] The proof combines a sharp additiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.24426","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/2606.24426/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":"2606.24426","created_at":"2026-06-24T01:15:30.102698+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.24426v1","created_at":"2026-06-24T01:15:30.102698+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.24426","created_at":"2026-06-24T01:15:30.102698+00:00"},{"alias_kind":"pith_short_12","alias_value":"V25NXU2UANCY","created_at":"2026-06-24T01:15:30.102698+00:00"},{"alias_kind":"pith_short_16","alias_value":"V25NXU2UANCYI2QT","created_at":"2026-06-24T01:15:30.102698+00:00"},{"alias_kind":"pith_short_8","alias_value":"V25NXU2U","created_at":"2026-06-24T01:15:30.102698+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2606.29284","citing_title":"A Tur\\'an Theorem for Cayley Graphs","ref_index":9,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS","json":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS.json","graph_json":"https://pith.science/api/pith-number/V25NXU2UANCYI2QTD5PWA7E4YS/graph.json","events_json":"https://pith.science/api/pith-number/V25NXU2UANCYI2QTD5PWA7E4YS/events.json","paper":"https://pith.science/paper/V25NXU2U"},"agent_actions":{"view_html":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS","download_json":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS.json","view_paper":"https://pith.science/paper/V25NXU2U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.24426&json=true","fetch_graph":"https://pith.science/api/pith-number/V25NXU2UANCYI2QTD5PWA7E4YS/graph.json","fetch_events":"https://pith.science/api/pith-number/V25NXU2UANCYI2QTD5PWA7E4YS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS/action/storage_attestation","attest_author":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS/action/author_attestation","sign_citation":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS/action/citation_signature","submit_replication":"https://pith.science/pith/V25NXU2UANCYI2QTD5PWA7E4YS/action/replication_record"}},"created_at":"2026-06-24T01:15:30.102698+00:00","updated_at":"2026-06-24T01:15:30.102698+00:00"}