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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1611.05746","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-17T15:56:33Z","cross_cats_sorted":[],"title_canon_sha256":"69eb3bfd64654f0ed55e6b6573c711856eb78ae59290c8907cc7955d0b53c99b","abstract_canon_sha256":"d3f0df8367758082f86fd7aea34bfe39e326ac59e85467a6d50e75ded0af597c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:33.090789Z","signature_b64":"o/reXfYklnXr49IXzpzNYokJD1btKR4pEi76AUravib4momrBuTQ/KPrsp0XhSjodgkNdyjuQvGShPo/13oCAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55a90c5a8f07188dbced41b4de635140bc48b9a34a88045f8c206d0c16d891be","last_reissued_at":"2026-05-17T23:57:33.090259Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:33.090259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\frac1{k^2}$. 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