{"paper":{"title":"A linear-time algorithm for finding a complete graph minor in a dense graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Bruce Reed, Daniel J. Harvey, David R. Wood, Gwena\\\"el Joret, Vida Dujmovi\\'c","submitted_at":"2012-02-13T04:46:01Z","abstract_excerpt":"Let g(t) be the minimum number such that every graph G with average degree d(G) \\geq g(t) contains a K_{t}-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) \\in \\Theta(t*sqrt{log t}). This article shows that for all fixed \\epsilon > 0 and fixed sufficiently large t \\geq t(\\epsilon), if d(G) \\geq (2+\\epsilon)g(t) then we can find this K_{t}-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) \\geq 2^{t-2}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.2624","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"}