{"paper":{"title":"On the Chudnovsky-Seymour-Sullivan Conjecture on Cycles in Triangle-free Digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Liu, Jian Shen, Kevin Chen, Sean Karson","submitted_at":"2009-09-14T05:41:00Z","abstract_excerpt":"For a simple digraph $G$ without directed triangles or digons, let $\\beta(G)$ be the size of the smallest subset $X \\subseteq E(G)$ such that $G\\setminus X$ has no directed cycles, and let $\\gamma(G)$ be the number of unordered pairs of nonadjacent vertices in $G$. In 2008, Chudnovsky, Seymour, and Sullivan showed that $\\beta (G) \\le \\gamma(G)$, and conjectured that $\\beta (G) \\le \\gamma(G)/2$. Recently, Dunkum, Hamburger, and P\\'or proved that $\\beta (G) \\le 0.88 \\gamma(G)$. In this note, we prove that $\\beta (G) \\le 0.8616 \\gamma(G)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.2468","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0909.2468/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}