{"paper":{"title":"The biased odd cycle game","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Asaf Ferber, Cory Palmer, Hong Liu, Mate Vizer, Michael Krivelevich, Roman Glebov, Tomas Valla","submitted_at":"2012-10-16T10:14:33Z","abstract_excerpt":"In this paper we consider biased Maker-Breaker games played on the edge set of a given graph $G$. We prove that for every $\\delta>0$ and large enough $n$, there exists a constant $k$ for which if $\\delta(G)\\geq \\delta n$ and $\\chi(G)\\geq k$, then Maker can build an odd cycle in the $(1:b)$ game for $b=O(\\frac{n}{\\log^2 n})$. We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, {\\L}uczak and R\\\"{o}dl: is it true that for any positive constants $t$ and $b$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4342","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"}