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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0702147","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-02-06T16:02:48Z","cross_cats_sorted":[],"title_canon_sha256":"1ebf67a28eb678abd68c94475ffc0bb56bdb15e4f2111a1213bf4308ddbd34e0","abstract_canon_sha256":"95e024d3caa94bf193b52a9c472d413604c22d76916f5c109dd5c9574324d27f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:44.365577Z","signature_b64":"56aob3pCXtGUlO18J//kUK78DM9U6wNh5P9AgjdNWCEkc5Pjv10bMwEivlhWVz8wBl/xqajsDzVrz9VnYvamDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"42b2c71df26c7908f095a5411a716743fdac17167a411d6795da1eb5429eea12","last_reissued_at":"2026-05-18T03:41:44.364710Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:44.364710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\gamma(G)$. 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