{"paper":{"title":"The Crossing Number of Semi-Pair-Shellable Drawings of Complete Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Lutz Oettershagen, Petra Mutzel","submitted_at":"2018-05-16T10:00:11Z","abstract_excerpt":"The Harary-Hill Conjecture states that for $n\\geq 3$ every drawing of $K_n$ has at least \\begin{align*}\n  H(n) := \\frac{1}{4}\\Big\\lfloor\\frac{n}{2}\\Big\\rfloor\\Big\\lfloor\\frac{n-1}{2}\\Big\\rfloor\\Big\\lfloor\\frac{n-2}{2}\\Big\\rfloor\\Big\\lfloor\\frac{n-3}{2}\\Big\\rfloor \\end{align*} crossings. In general the problem remains unsolved, however there has been some success in proving the conjecture for restricted classes of drawings. The most recent and most general of these classes is seq-shellability. In this work, we improve these results and introduce the new class of semi-pair-shellable drawings. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.06780","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"}