{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:2BMK6NIHV7CTERZPOUSUGDRLLD","short_pith_number":"pith:2BMK6NIH","schema_version":"1.0","canonical_sha256":"d058af3507afc532472f7525430e2b58ef57007021804bb2e140a096975e3455","source":{"kind":"arxiv","id":"1903.03908","version":1},"attestation_state":"computed","paper":{"title":"Asymptotic Density of Graphs Excluding Disconnected Minors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rohan Kapadia, Sergey Norin, Yingjie Qian","submitted_at":"2019-03-10T02:58:46Z","abstract_excerpt":"For a graph $H$, let $$c_{\\infty}(H)= \\lim_{n \\to \\infty}\\max\\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\\infty}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\\infty}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\\infty}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_\\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint copies of 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":"1903.03908","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-10T02:58:46Z","cross_cats_sorted":[],"title_canon_sha256":"d724a236cde18436283a2705b0efe89cba30c2ca2fb01ead34c184b017282cfa","abstract_canon_sha256":"13ac38312c72a565a29b4d2f26214c8f6d0754715879e06123f045889cf3bbce"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:40.623878Z","signature_b64":"vx18EpZD3BQUnig9p76CN7irp/XTGlU8UGKzX/866dJrxAzDzmp4E5K9VO/nNRoMLvJ60+RlaOm7y8xlk380Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d058af3507afc532472f7525430e2b58ef57007021804bb2e140a096975e3455","last_reissued_at":"2026-05-17T23:51:40.623267Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:40.623267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic Density of Graphs Excluding Disconnected Minors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rohan Kapadia, Sergey Norin, Yingjie Qian","submitted_at":"2019-03-10T02:58:46Z","abstract_excerpt":"For a graph $H$, let $$c_{\\infty}(H)= \\lim_{n \\to \\infty}\\max\\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\\infty}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\\infty}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\\infty}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_\\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint copies of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03908","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":"1903.03908","created_at":"2026-05-17T23:51:40.623368+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.03908v1","created_at":"2026-05-17T23:51:40.623368+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.03908","created_at":"2026-05-17T23:51:40.623368+00:00"},{"alias_kind":"pith_short_12","alias_value":"2BMK6NIHV7CT","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"2BMK6NIHV7CTERZP","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"2BMK6NIH","created_at":"2026-05-18T12:33:07.085635+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/2BMK6NIHV7CTERZPOUSUGDRLLD","json":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD.json","graph_json":"https://pith.science/api/pith-number/2BMK6NIHV7CTERZPOUSUGDRLLD/graph.json","events_json":"https://pith.science/api/pith-number/2BMK6NIHV7CTERZPOUSUGDRLLD/events.json","paper":"https://pith.science/paper/2BMK6NIH"},"agent_actions":{"view_html":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD","download_json":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD.json","view_paper":"https://pith.science/paper/2BMK6NIH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.03908&json=true","fetch_graph":"https://pith.science/api/pith-number/2BMK6NIHV7CTERZPOUSUGDRLLD/graph.json","fetch_events":"https://pith.science/api/pith-number/2BMK6NIHV7CTERZPOUSUGDRLLD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD/action/storage_attestation","attest_author":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD/action/author_attestation","sign_citation":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD/action/citation_signature","submit_replication":"https://pith.science/pith/2BMK6NIHV7CTERZPOUSUGDRLLD/action/replication_record"}},"created_at":"2026-05-17T23:51:40.623368+00:00","updated_at":"2026-05-17T23:51:40.623368+00:00"}