{"paper":{"title":"On topological graphs with at most four crossings per edge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Eyal Ackerman","submitted_at":"2015-09-07T07:24:05Z","abstract_excerpt":"We show that if a graph $G$ with $n \\geq 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\\v{c}i\\'{c}, Tardos, and T\\'oth, and yields a better bound for the famous Crossing Lemma: The crossing number, $\\mbox{cr}(G)$, of a (not too sparse) graph $G$ with $n$ vertices and $m$ edges is at least $c\\frac{m^3}{n^2}$, where $c > 1/29$. This bound is known to be tight, apart from the constant $c$ for which the previous best lower bound was $1/31.1$. As another corollary w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01932","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"}