{"paper":{"title":"Two (Known) Results About Graphs with No Short Odd Cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Saladi Rahul, Sariel Har-Peled","submitted_at":"2018-10-03T16:31:24Z","abstract_excerpt":"Consider a graph with $n$ vertices where the shortest odd cycle is of length $>2k+1$. We revisit two known results about such graphs:\n  (I) Such a graph is almost bipartite, in the sense that it can be made bipartite by removing from it $O\\bigl( (n/k) \\log (n/k) \\bigr)$ vertices. While this result is known [GKL97] -- our new proof seems to yield slightly better constants, and is (arguably) conceptually simpler. To this end, we state (and prove) a version of CKR partitions [CKR01, FRT04] that has a small vertex separator, and it might be of independent interest. While this must be known in the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.01832","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":""},"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"}