{"paper":{"title":"Strong orientation of a connected graph for a crossing family","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ahmad Abdi, Mahsa Dalirrooyfard, Meike Neuwohner","submitted_at":"2024-11-20T11:04:48Z","abstract_excerpt":"Given a connected graph $G=(V,E)$ and a crossing family $\\mathcal{C}$ over ground set $V$ such that $|\\delta_G(U)|\\geq 2$ for every $U\\in \\mathcal{C}$, we prove there exists a strong orientation of $G$ for $\\mathcal{C}$, i.e., an orientation of $G$ such that each set in $\\mathcal{C}$ has at least one outgoing and at least one incoming arc. This implies the main conjecture in Chudnovsky et al. (Disjoint dijoins. Journal of Combinatorial Theory, Series B, 120:18--35, 2016). In particular, in every minimal counterexample to the Edmonds-Giles conjecture where the minimum weight of a dicut is $2$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.13202","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/2411.13202/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"}