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For $k=3$, the bound of $6n-12$ has been improved to $\\frac{11}{2}n-11$ and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of $\\frac{11}{2}n-11$ edges also holds for non-simple $3$-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are"},"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":"1602.04995","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2016-02-16T11:45:25Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"39de28c4f3fe23d519a0ddd249437b7ad9add93aa7b0bc6f01d958d8f6d81237","abstract_canon_sha256":"2a1fdd12118935a13ad75c17c36aa3c7828972c4016ed86ea1dc3836412fb40f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:20.861935Z","signature_b64":"30+SXrv2uB81CLGGO01B8MtMOly+8Xwf5gdzFJfKeLNDeQvWMl5TxG4pPltXfFCyFNpdgN3lS06ciYLyju+wDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"13bb107df3dd7803c6f3c0785846b06a7737a31bc2193526d8edb694b73d4ab7","last_reissued_at":"2026-05-18T01:07:20.861462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:20.861462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Density of non-Simple 3-Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CG","authors_text":"Chrysanthi N. 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