{"paper":{"title":"Generalizations and strengthenings of Ryser's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Grace McCourt, Hannah Sheats, Louis DeBiasio, Yigal Kamel","submitted_at":"2020-09-15T17:24:54Z","abstract_excerpt":"Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\\nu(H)$, the vertex cover number is at most $(r-1)\\nu(H)$. This far reaching generalization of K\\\"onig's theorem is only known to be true for $r\\leq 3$, or $\\nu(G)=1$ and $r\\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\\alpha(G)$, there exists at most $(r-1)\\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$.\n  We make the case that this latter formulation of Ryser's conjecture naturally leads t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.07239","kind":"arxiv","version":3},"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/2009.07239/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"}