{"paper":{"title":"Shellable drawings and the cylindrical crossing number of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Bernardo M. \\'Abrego, Gelasio Salazar, Oswin Aichholzer, Pedro Ramos, Silvia Fern\\'andez-Merchant","submitted_at":"2013-09-14T13:24:20Z","abstract_excerpt":"The Harary-Hill Conjecture States that the number of crossings in any drawing of the complete graph $ K_n $ in the plane is at least $Z(n):=\\frac{1}{4}\\left\\lfloor \\frac{n}{2}\\right\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor \\frac{n-2}{2}\\right\\rfloor\\left\\lfloor \\frac{n-3}{2}\\right\\rfloor$. In this paper, we settle the Harary-Hill conjecture for {\\em shellable drawings}. We say that a drawing $D$ of $ K_n $ is {\\em $ s $-shellable} if there exist a subset $ S = \\{v_1,v_2,\\ldots,v_ s\\}$ of the vertices and a region $R$ of $D$ with the following property: For all $1 \\leq i < j \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3665","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"}