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Since Kleitman's verification of ZC for $K_{5,n}$ (from which ZC for $K_{6,n}$ easily follows), very little progress has been made around ZC; the most notable exceptions involve computer-aided results. With the aim of gaining a more profound understanding of this notoriously difficult conjecture, we investigate the optimal (that is, crossing-minimal) drawings of $K_{5,n}$. The widely known natural drawings of $K_{m,n}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1210.1988","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-06T19:14:39Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"fb2b70611175d238d62fa004685ba4fd7a55e33d914e24c43bdbcc9c60fa7ecc","abstract_canon_sha256":"6c4ea3638e1aebf1f6a43a5c71773b6a6997bf8a7e86bfb81af229b1485a6006"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:44.692860Z","signature_b64":"zWMQnSJRrlGX4KQ8YwM+FZMea4axkudEBE28KpTzdFbArr84LEARjfwGmxini08cUnH59lJcHqC9Evn4TRwxBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0183aa6d90dae6f31635e2dcb492aa4476805825504bd673c09d33296fde9086","last_reissued_at":"2026-05-17T23:47:44.692433Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:44.692433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The optimal drawings of K_{5,n}","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Carolina Medina, Cesar Hernandez-Velez, Gelasio Salazar","submitted_at":"2012-10-06T19:14:39Z","abstract_excerpt":"Zarankiewicz's Conjecture (ZC) states that the crossing number cr$(K_{m,n})$ equals $Z(m,n):=\\floor{\\frac{m}{2}} \\floor{\\frac{m-1}{2}} \\floor{\\frac{n}{2}} \\floor{\\frac{n-1}{2}}$. Since Kleitman's verification of ZC for $K_{5,n}$ (from which ZC for $K_{6,n}$ easily follows), very little progress has been made around ZC; the most notable exceptions involve computer-aided results. With the aim of gaining a more profound understanding of this notoriously difficult conjecture, we investigate the optimal (that is, crossing-minimal) drawings of $K_{5,n}$. 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