{"paper":{"title":"Using Block Designs in Crossing Number Bounds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Arran Hamm, Eva Czabarka, Garner Cochran, Gregory Clark, Gwen Spencer, John Asplund, Laszlo Szekely, Libby Taylor, Zhiyu Wang","submitted_at":"2018-07-10T00:41:13Z","abstract_excerpt":"The crossing number ${\\mbox {cr}}(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\\ge 1$, the $k$-planar crossing number of $G$, ${\\mbox {cr}}_k(G)$, is defined as the minimum of ${\\mbox {cr}}(G_1)+{\\mbox {cr}}(G_2)+\\ldots+{\\mbox {cr}}(G_{k})$ over all graphs $G_1, G_2,\\ldots, G_{k}$ with $\\cup_{i=1}^{k}G_i=G$. Pach et al. [\\emph{Computational Geometry: Theory and Applications} {\\bf 68} 2--6, (2018)] showed that for every $k\\ge 1$, we have ${\\mbox {cr}}_k(G)\\le \\left(\\frac{2}{k^2}-\\frac1{k^3}\\right){\\mbox {cr}}(G)$ and that thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03430","kind":"arxiv","version":1},"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"}