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Setting $\\omega(\\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka."},"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":"1102.1023","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-02-04T21:48:30Z","cross_cats_sorted":[],"title_canon_sha256":"b3af2702bade2c355e90fd5bc49fb262dd0e4a93faa850eced6c6b6b49db2ba3","abstract_canon_sha256":"756ab4220ecf44bf140ceb10dc8c1adc1b7e4a6b034dd510db13c8d6a8719df8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:55.176062Z","signature_b64":"HikmJNW5TGCHu7oVQK98StijdrEJaya5Xq0hGPRHJjU8vf6cK7GVMW6brvFzTAhLVh5rPR6VkXJ5p/Y+g+vVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"061b4e8673336ae57932b6476ced9aa0134716a6c788dcbf868cd31cfe2717fb","last_reissued_at":"2026-05-18T04:29:55.175559Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:55.175559Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coloring $\\Delta$-Critical Graphs With Small High Vertex Cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Landon Rabern","submitted_at":"2011-02-04T21:48:30Z","abstract_excerpt":"We prove that $K_{\\chi(G)}$ is the only critical graph $G$ with $\\chi(G) \\geq \\Delta(G) \\geq 6$ and $\\omega(\\mathcal{H}(G)) \\leq \\left \\lfloor \\frac{\\Delta(G)}{2} \\right \\rfloor - 2$. Here $\\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $\\chi(G)$. Setting $\\omega(\\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1023","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":"1102.1023","created_at":"2026-05-18T04:29:55.175637+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.1023v1","created_at":"2026-05-18T04:29:55.175637+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.1023","created_at":"2026-05-18T04:29:55.175637+00:00"},{"alias_kind":"pith_short_12","alias_value":"AYNU5BTTGNVO","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"AYNU5BTTGNVOK6JS","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"AYNU5BTT","created_at":"2026-05-18T12:26:24.575870+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/AYNU5BTTGNVOK6JSWZDWZ3M2UA","json":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA.json","graph_json":"https://pith.science/api/pith-number/AYNU5BTTGNVOK6JSWZDWZ3M2UA/graph.json","events_json":"https://pith.science/api/pith-number/AYNU5BTTGNVOK6JSWZDWZ3M2UA/events.json","paper":"https://pith.science/paper/AYNU5BTT"},"agent_actions":{"view_html":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA","download_json":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA.json","view_paper":"https://pith.science/paper/AYNU5BTT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.1023&json=true","fetch_graph":"https://pith.science/api/pith-number/AYNU5BTTGNVOK6JSWZDWZ3M2UA/graph.json","fetch_events":"https://pith.science/api/pith-number/AYNU5BTTGNVOK6JSWZDWZ3M2UA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA/action/storage_attestation","attest_author":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA/action/author_attestation","sign_citation":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA/action/citation_signature","submit_replication":"https://pith.science/pith/AYNU5BTTGNVOK6JSWZDWZ3M2UA/action/replication_record"}},"created_at":"2026-05-18T04:29:55.175637+00:00","updated_at":"2026-05-18T04:29:55.175637+00:00"}