{"paper":{"title":"Point sets that minimize $(\\le k)$-edges, 3-decomposable drawings, and the rectilinear crossing number of $K_{30}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C. Hern\\'andez-V\\'elez, C. Villalobos, J. Lea\\~nos, M. Cetina","submitted_at":"2010-09-23T23:07:53Z","abstract_excerpt":"There are two properties shared by all known crossing-minimizing geometric drawings of $K_n$, for $n$ a multiple of 3. First, the underlying $n$-point set of these drawings has exactly $3\\binom{k+2}{2}$ $(\\le k)$-edges, for all $0\\le k < n/3$. Second, all such drawings have the $n$ points divided into three groups of equal size; this last property is captured under the concept of 3-decomposability. In this paper we show that these properties are tightly related: every $n$-point set with exactly $3\\binom{k+2}{2}$ $(\\le k)$-edges for all $0\\le k < n/3$, is 3-decomposable. As an application, we p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.4736","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"}