{"paper":{"title":"On subgraphs of $C_{2k}$-free graphs and a problem of K\\\"uhn and Osthus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abhishek Methuku, Casey Tompkins, D\\'aniel Gr\\'osz","submitted_at":"2017-08-17T22:43:55Z","abstract_excerpt":"Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\\H{o}ri et al. showed that $\\frac{3}{8} \\le c \\le \\frac{2}{5}$. We prove that $c=\\frac{3}{8}$. More generally, we show that for any $\\varepsilon>0$, and any integer $k \\ge 2$, there is a $C_{2k}$-free graph $G_1$ which does not contain a bipartite subgraph of girth greater than $2k$ with more than $\\left(1-\\frac{1}{2^{2k-2}}\\right)\\frac{2}{2k-1}(1+\\varepsilon)$ fraction of the edges of $G_1$. There also exists a $C_{2k}$-free graph $G_2$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05454","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":""},"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"}