{"paper":{"title":"Transversals in a collections of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ethan Y.H. Li, Luyi Li, Ping Li","submitted_at":"2023-10-10T06:43:57Z","abstract_excerpt":"Let $\\mathcal{S}$ be a fixed family of graphs on vertex set $V$ and $\\mathcal{G}$ be a collection of elements in $\\mathcal{S}$. We investigated the transversal problem of finding the maximum value of $|\\mathcal{G}|$ when $\\mathcal{G}$ contains no rainbow elements in $\\mathcal{S}$. Specifically, we determine the exact values when $\\mathcal{S}$ is a family of stars or a family of trees of the same order $n$ with $n$ dividing $|V|$. Further, all the extremal cases for $\\mathcal{G}$ are characterized."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2310.06354","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/2310.06354/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"}