{"paper":{"title":"Note on k-planar crossing numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Csaba D. T\\'oth, G\\'eza T\\'oth, J\\'anos Pach, L\\'aszl\\'o A. Sz\\'ekely","submitted_at":"2016-11-17T15:56:33Z","abstract_excerpt":"The crossing number $cr(G)$ 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 $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\\ldots, G_{k-1}$ with $\\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\\ge 1$, we have $cr_k(G)\\le \\left(\\frac{2}{k^2}-\\frac1{k^3}\\right)cr(G)$. This bound does not remain true if we replace the constant $\\frac{2}{k^2}-\\frac1{k^3}$ by any number smaller than $\\frac1{k^2}$. Some of the results extend to th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05746","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"}