{"paper":{"title":"A note on independence number, connectivity and $k$-ended tree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Pham Hoang Ha","submitted_at":"2018-10-26T10:01:37Z","abstract_excerpt":"A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\\geq 2$). Let $S$ be a vertex subset of $G$ such that $\\alpha_{G}(S) \\leq k + \\kappa_{G}(S)- 1.$ Then, $G$ has a $k$-ended tree which covers $S.$ Moreover, the condition is sharp."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.11242","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":""},"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"}