{"paper":{"title":"Cycles in dense digraphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Blair D. Sullivan, Maria Chudnovsky, Paul Seymour","submitted_at":"2007-02-06T16:02:48Z","abstract_excerpt":"Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let $\\beta(G)$ denote the size of the smallest subset X in E(G) such that $G\\X$ has no directed cycles, and let $\\gamma(G)$ be the number of unordered pairs {u,v} of vertices such that u,v are nonadjacent in G. It is easy to see that if $\\gamma(G) = 0$ then $\\beta(G) = 0$; what can we say about $\\beta(G)$ if $\\gamma(G)$ is bounded?\n  We prove that in general $\\beta(G)$ is at most $\\gamma(G)$. We conjecture that in fact $\\beta(G)$ is at most $\\gamma(G)/2$ (this would be best possible if true), a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702147","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"}