{"paper":{"title":"Crossing numbers and combinatorial characterization of monotone drawings of $K_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Kyn\\v{c}l, Martin Balko, Radoslav Fulek","submitted_at":"2013-12-13T00:00:39Z","abstract_excerpt":"In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = \\frac{1}{4} \\lfloor\\frac{n}{2}\\rfloor \\left\\lfloor\\frac{n-1}{2}\\right\\rfloor \\left\\lfloor\\frac{n-2}{2}\\right\\rfloor\\left\\lfloor\\frac{n-3}{2}\\right\\rfloor$. Generalizing the result by \\'{A}brego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.3679","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"}