{"paper":{"title":"Dirac's theorem for linear hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hyunwoo Lee, Seonghyuk Im","submitted_at":"2024-03-21T10:10:32Z","abstract_excerpt":"Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\\delta(G) \\geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\\frac{n}{k} + \\Omega(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining a perfect matching is impossible with any degree condition. Furthermore, we show that if $\\delta(H) \\geq (\\frac{1}{k}+o(1))n$, then $H$ contains almos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.14269","kind":"arxiv","version":2},"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/2403.14269/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"}