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We show that (2 \\alpha(G) - \\lceil log_t(t \\alpha(G)/2) \\rceil) h(G) \\geq |V(G)| where t is approximately 6.83. For graphs with \\alpha(G) \\geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 \\alpha(G) - 2) h(G) >= |V(G)| when \\alpha(G) >= 3."},"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":"0907.2421","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-07-14T18:17:25Z","cross_cats_sorted":[],"title_canon_sha256":"6fec8069943cb87e05e82848c89b9d889212281334028ba7d8853056cdf0f968","abstract_canon_sha256":"0f14695573ef686d34f2bdc382935aea7273662e8b7072a83861a9d4620692df"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:07.456963Z","signature_b64":"FEn1iMIR6Am7aHs/tkDFm5CK5i8Livpouv0OsBLBrxAZQTzgV1Va4ZPfvkaT8y2kdToxkHXN2o2YSk7X1MGJDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15e77bb826ffefa37d591a2925a8a84bb26b7f1b5392c4cce02a9ff015e86ee4","last_reissued_at":"2026-05-18T04:11:07.456393Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:07.456393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Complete Minors, Independent Sets, and Chordal Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hehui Wu, John Lenz, Jozsef Balogh","submitted_at":"2009-07-14T18:17:25Z","abstract_excerpt":"The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. 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For graphs with \\alpha(G) \\geq 14, this improves on a recent result of Kawarabayashi and Song who showed (2 \\alpha(G) - 2) h(G) >= |V(G)| when \\alpha(G) >= 3."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2421","kind":"arxiv","version":2},"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":"0907.2421","created_at":"2026-05-18T04:11:07.456475+00:00"},{"alias_kind":"arxiv_version","alias_value":"0907.2421v2","created_at":"2026-05-18T04:11:07.456475+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2421","created_at":"2026-05-18T04:11:07.456475+00:00"},{"alias_kind":"pith_short_12","alias_value":"CXTXXOBG77X2","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"CXTXXOBG77X2G7KZ","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"CXTXXOBG","created_at":"2026-05-18T12:25:59.703012+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/CXTXXOBG77X2G7KZDIUSLKFIJO","json":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO.json","graph_json":"https://pith.science/api/pith-number/CXTXXOBG77X2G7KZDIUSLKFIJO/graph.json","events_json":"https://pith.science/api/pith-number/CXTXXOBG77X2G7KZDIUSLKFIJO/events.json","paper":"https://pith.science/paper/CXTXXOBG"},"agent_actions":{"view_html":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO","download_json":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO.json","view_paper":"https://pith.science/paper/CXTXXOBG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0907.2421&json=true","fetch_graph":"https://pith.science/api/pith-number/CXTXXOBG77X2G7KZDIUSLKFIJO/graph.json","fetch_events":"https://pith.science/api/pith-number/CXTXXOBG77X2G7KZDIUSLKFIJO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO/action/storage_attestation","attest_author":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO/action/author_attestation","sign_citation":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO/action/citation_signature","submit_replication":"https://pith.science/pith/CXTXXOBG77X2G7KZDIUSLKFIJO/action/replication_record"}},"created_at":"2026-05-18T04:11:07.456475+00:00","updated_at":"2026-05-18T04:11:07.456475+00:00"}