{"paper":{"title":"Longest odd cycles in non-bipartite $C_{2k+1}$-free graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rui Wang, Shipeng Wang","submitted_at":"2025-08-22T08:18:58Z","abstract_excerpt":"In strengthening a result of Andr\\'asfai, Erd\\H{o}s and S\\'os in 1974, H\\\"{a}ggkvist proved that if $G$ is an $n$-vertex $C_{2k+1}$-free graph with minimum degree $\\delta(G)>\\frac{2n}{2k+3}$ and $n>\\binom{k+2}{2}(2k+3)(3k+2)$, then $G$ contains no odd cycle of length greater than $\\frac{k+1}{2}$. This result has many applications.In this paper, we consider a similar problem by replacing minimum degree condition with edge number condition. We prove that for integers $n,k,r$ with $k\\geq 2,3\\leq r\\leq 2k$ and $n \\geq 2\\left(r+2\\right)\\left(r+1\\right)\\left(r+2k\\right)$, if $G$ is an $n$-vertex $C_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.16199","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/2508.16199/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"}