{"paper":{"title":"Complete Minors, Independent Sets, and Chordal Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hehui Wu, John Lenz, Jozsef Balogh","submitted_at":"2009-07-14T18:17:25Z","abstract_excerpt":"The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger's Conjecture states that h(G) >= \\chi(G). Since \\chi(G) \\alpha(G) >= |V(G)|, Hadwiger's Conjecture implies that \\alpha(G) h(G) >= |V(G)|. We show that (2 \\alpha(G) - \\lceil log_t(t \\alpha(G)/2) \\rceil) h(G) \\geq |V(G)| where t is approximately 6.83. For graphs with \\alpha(G) \\geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 \\alpha(G) - 2) h(G) >= |V(G)| when \\alpha(G) >= 3."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2421","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":""},"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"}