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Brandenburg et al. showed that there are maximal 1-planar graphs with only $\\frac{45}{17}n + O(1)\\approx 2.647n$ edges and maximal 1-plane graphs with only $\\frac{7}{3}n+O(1)\\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\\frac{28}{13}n-O(1)\\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges.\n  We improve both lower bounds to $\\frac{20n}{9}\\approx 2.22n$."},"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":"1509.05548","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-09-18T08:50:56Z","cross_cats_sorted":[],"title_canon_sha256":"95374c80053e06caf6aa97db2f7071f45fedf3dbef0c18deb680351c3fe33c28","abstract_canon_sha256":"a6c79cd1cd446ca2c13f0d3b5231b5cf662c285447a61703fcf9c63ee2f6baf6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.869969Z","signature_b64":"F81IkDaMJ7jNZctJDVGWLvBTxB60fYuSCy8NFQgvM+8yHJ/GicE+PLkQAjWF8iccOTkc6vCpjoyOumOpJvjbDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0af4a841d596726a6109888ed9c91e48a315cc97c410f7fc8ecf0a74729a68b3","last_reissued_at":"2026-05-18T01:32:42.869519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.869519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improvements on the density of maximal 1-planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'eza T\\'oth, J\\'anos Bar\\'at","submitted_at":"2015-09-18T08:50:56Z","abstract_excerpt":"A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\\frac{45}{17}n + O(1)\\approx 2.647n$ edges and maximal 1-plane graphs with only $\\frac{7}{3}n+O(1)\\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\\frac{28}{13}n-O(1)\\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges.\n  We improve both lower bounds to $\\frac{20n}{9}\\approx 2.22n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.05548","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.05548","created_at":"2026-05-18T01:32:42.869608+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.05548v1","created_at":"2026-05-18T01:32:42.869608+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.05548","created_at":"2026-05-18T01:32:42.869608+00:00"},{"alias_kind":"pith_short_12","alias_value":"BL2KQQOVSZZG","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BL2KQQOVSZZGUYIJ","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BL2KQQOV","created_at":"2026-05-18T12:29:14.074870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC","json":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC.json","graph_json":"https://pith.science/api/pith-number/BL2KQQOVSZZGUYIJRCHNTSI6JC/graph.json","events_json":"https://pith.science/api/pith-number/BL2KQQOVSZZGUYIJRCHNTSI6JC/events.json","paper":"https://pith.science/paper/BL2KQQOV"},"agent_actions":{"view_html":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC","download_json":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC.json","view_paper":"https://pith.science/paper/BL2KQQOV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.05548&json=true","fetch_graph":"https://pith.science/api/pith-number/BL2KQQOVSZZGUYIJRCHNTSI6JC/graph.json","fetch_events":"https://pith.science/api/pith-number/BL2KQQOVSZZGUYIJRCHNTSI6JC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC/action/storage_attestation","attest_author":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC/action/author_attestation","sign_citation":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC/action/citation_signature","submit_replication":"https://pith.science/pith/BL2KQQOVSZZGUYIJRCHNTSI6JC/action/replication_record"}},"created_at":"2026-05-18T01:32:42.869608+00:00","updated_at":"2026-05-18T01:32:42.869608+00:00"}