{"paper":{"title":"Bishellable drawings 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, Birgit Vogtenhuber, Bojan Mohar, Dan McQuillan, Oswin Aichholzer, Pedro Ramos, Petra Mutzel, R. Bruce Richter, Silvia Fern\\'andez-Merchant","submitted_at":"2015-10-02T10:23:19Z","abstract_excerpt":"The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \\frac 1 4 \\left\\lfloor\\frac{\\mathstrut n}{\\mathstrut 2}\\right\\rfloor \\left\\lfloor\\frac{\\mathstrut n-1}{\\mathstrut 2}\\right\\rfloor \\left\\lfloor\\frac{\\mathstrut n-2}{\\mathstrut 2}\\right\\rfloor \\left\\lfloor\\frac{\\mathstrut n-3}{\\mathstrut 2}\\right \\rfloor$. \\'Abrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\\geq\\lfloor\\frac{n}{2}\\rfloor$, then $D$ has at least $H(n)$ crossings. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00549","kind":"arxiv","version":3},"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"}