{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:2KIODM6CJGIKZK3MVJ4ATG3FS3","short_pith_number":"pith:2KIODM6C","schema_version":"1.0","canonical_sha256":"d290e1b3c24990acab6caa78099b6596c5ba43b1c719632e2d6da361a8c60703","source":{"kind":"arxiv","id":"2408.16085","version":1},"attestation_state":"computed","paper":{"title":"On $k$-planar Graphs without Short Cycles","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Aaron B\\\"ungener, Alexandra Weinberger, Michael A. Bekos, Michael Hoffmann, Michael Kaufmann, Pat Morin, Prosenjit Bose, Saeed Odak, Vida Dujmovi\\'c","submitted_at":"2024-08-28T18:28:23Z","abstract_excerpt":"We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\\ge 5$). For all three settings and all $k\\in\\{1,2,3\\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. Fo"},"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":"2408.16085","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-28T18:28:23Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3fc9f5c808ea0ed99352f31c292022ce6c1924e74db33122a0dbbccd5f48a872","abstract_canon_sha256":"cf4df3ff624fa5f1c8ec84455bd0528a1c8894df3e2d5b1f518be86a75a6b705"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:00:25.746908Z","signature_b64":"L4XwdQ3jMSjz95r9ddraPp2vAjKANSXjFwkNq35y8fraikfDQQhbhEW0AIz12xZMPOHWjOTL0jfCuW5CCQ6nDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d290e1b3c24990acab6caa78099b6596c5ba43b1c719632e2d6da361a8c60703","last_reissued_at":"2026-07-05T09:00:25.746392Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:00:25.746392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On $k$-planar Graphs without Short Cycles","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Aaron B\\\"ungener, Alexandra Weinberger, Michael A. Bekos, Michael Hoffmann, Michael Kaufmann, Pat Morin, Prosenjit Bose, Saeed Odak, Vida Dujmovi\\'c","submitted_at":"2024-08-28T18:28:23Z","abstract_excerpt":"We study the impact of forbidding short cycles to the edge density of $k$-planar graphs; a $k$-planar graph is one that can be drawn in the plane with at most $k$ crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are $3$-cycles, $4$-cycles or both of them (i.e., girth $\\ge 5$). For all three settings and all $k\\in\\{1,2,3\\}$, we present lower and upper bounds on the maximum number of edges in any $k$-planar graph on $n$ vertices. Our bounds are of the form $c\\,n$, for some explicit constant $c$ that depends on $k$ and on the setting. Fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.16085","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.16085/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2408.16085","created_at":"2026-07-05T09:00:25.746469+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.16085v1","created_at":"2026-07-05T09:00:25.746469+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.16085","created_at":"2026-07-05T09:00:25.746469+00:00"},{"alias_kind":"pith_short_12","alias_value":"2KIODM6CJGIK","created_at":"2026-07-05T09:00:25.746469+00:00"},{"alias_kind":"pith_short_16","alias_value":"2KIODM6CJGIKZK3M","created_at":"2026-07-05T09:00:25.746469+00:00"},{"alias_kind":"pith_short_8","alias_value":"2KIODM6C","created_at":"2026-07-05T09:00:25.746469+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.00878","citing_title":"The minimum size of maximal bipartite IC-plane graphs with given connectivity","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3","json":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3.json","graph_json":"https://pith.science/api/pith-number/2KIODM6CJGIKZK3MVJ4ATG3FS3/graph.json","events_json":"https://pith.science/api/pith-number/2KIODM6CJGIKZK3MVJ4ATG3FS3/events.json","paper":"https://pith.science/paper/2KIODM6C"},"agent_actions":{"view_html":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3","download_json":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3.json","view_paper":"https://pith.science/paper/2KIODM6C","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.16085&json=true","fetch_graph":"https://pith.science/api/pith-number/2KIODM6CJGIKZK3MVJ4ATG3FS3/graph.json","fetch_events":"https://pith.science/api/pith-number/2KIODM6CJGIKZK3MVJ4ATG3FS3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3/action/storage_attestation","attest_author":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3/action/author_attestation","sign_citation":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3/action/citation_signature","submit_replication":"https://pith.science/pith/2KIODM6CJGIKZK3MVJ4ATG3FS3/action/replication_record"}},"created_at":"2026-07-05T09:00:25.746469+00:00","updated_at":"2026-07-05T09:00:25.746469+00:00"}