{"paper":{"title":"A strong structural stability of $C_{2k+1}$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yuejian Peng, Zilong Yan","submitted_at":"2024-08-28T02:21:37Z","abstract_excerpt":"F\\\"uredi and Gunderson showed that $ex(n, C_{2k+1})$ is achieved only on $K_{\\lfloor\\frac{n}{2}\\rfloor, \\lceil\\frac{n}{2}\\rceil}$ if $n\\ge 4k-2$. It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\\lfloor\\frac{n-r+1}{2}\\rfloor, \\lceil\\frac{n-r+1}{2}\\rceil}$. We show that for integers $r, k$ with $3\\le r\\le 2k-4$ and $n\\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.15487","kind":"arxiv","version":2},"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/2408.15487/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"}