{"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"}