{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:TVU6FTYWOSVK6TMY53EZBPUSX5","short_pith_number":"pith:TVU6FTYW","schema_version":"1.0","canonical_sha256":"9d69e2cf1674aaaf4d98eec990be92bf5d0fd86a95286cf99e4192ab6cf9c88a","source":{"kind":"arxiv","id":"1511.07591","version":1},"attestation_state":"computed","paper":{"title":"Almost All Regular Graphs are Normal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Seyed Saeed Changiz Rezaei, Seyyed Aliasghar Hosseini","submitted_at":"2015-11-24T06:50:24Z","abstract_excerpt":"In 1999, De Simone and K\\\"{o}rner conjectured that every graph without induced $C_5,C_7,\\overline{C}_7$ contains a clique cover $\\mathcal C$ and a stable set cover $\\mathcal I$ such that every clique in $\\mathcal C$ and every stable set in $\\mathcal I$ have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This implies that for every fixed $d$, random $d$-regular graphs are a.a.s. normal."},"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":"1511.07591","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-24T06:50:24Z","cross_cats_sorted":[],"title_canon_sha256":"60a8a5374153182ba60caa98c8940859e3d339d9db6462f761ec298e6619b666","abstract_canon_sha256":"91aa7727acc5c8c50b09526a59ed6a0efa0bbfe459030435df4c7d465262df6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:04.314143Z","signature_b64":"casKrptO9TSJpAxVI4P21O8TggmM0SyclKyGZW/L8ppYYzf2TDmX7dg3Fg+0dnRqk/VIy8q3ytJjSKPYO263AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d69e2cf1674aaaf4d98eec990be92bf5d0fd86a95286cf99e4192ab6cf9c88a","last_reissued_at":"2026-05-18T01:26:04.313511Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:04.313511Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost All Regular Graphs are Normal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bojan Mohar, Seyed Saeed Changiz Rezaei, Seyyed Aliasghar Hosseini","submitted_at":"2015-11-24T06:50:24Z","abstract_excerpt":"In 1999, De Simone and K\\\"{o}rner conjectured that every graph without induced $C_5,C_7,\\overline{C}_7$ contains a clique cover $\\mathcal C$ and a stable set cover $\\mathcal I$ such that every clique in $\\mathcal C$ and every stable set in $\\mathcal I$ have a vertex in common. This conjecture has roots in information theory and became known as the Normal Graph Conjecture. Here we prove that all graphs of bounded maximum degree and sufficiently large odd girth (linear in the maximum degree) are normal. This implies that for every fixed $d$, random $d$-regular graphs are a.a.s. normal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07591","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":"1511.07591","created_at":"2026-05-18T01:26:04.313626+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.07591v1","created_at":"2026-05-18T01:26:04.313626+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07591","created_at":"2026-05-18T01:26:04.313626+00:00"},{"alias_kind":"pith_short_12","alias_value":"TVU6FTYWOSVK","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TVU6FTYWOSVK6TMY","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TVU6FTYW","created_at":"2026-05-18T12:29:42.218222+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/TVU6FTYWOSVK6TMY53EZBPUSX5","json":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5.json","graph_json":"https://pith.science/api/pith-number/TVU6FTYWOSVK6TMY53EZBPUSX5/graph.json","events_json":"https://pith.science/api/pith-number/TVU6FTYWOSVK6TMY53EZBPUSX5/events.json","paper":"https://pith.science/paper/TVU6FTYW"},"agent_actions":{"view_html":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5","download_json":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5.json","view_paper":"https://pith.science/paper/TVU6FTYW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.07591&json=true","fetch_graph":"https://pith.science/api/pith-number/TVU6FTYWOSVK6TMY53EZBPUSX5/graph.json","fetch_events":"https://pith.science/api/pith-number/TVU6FTYWOSVK6TMY53EZBPUSX5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5/action/storage_attestation","attest_author":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5/action/author_attestation","sign_citation":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5/action/citation_signature","submit_replication":"https://pith.science/pith/TVU6FTYWOSVK6TMY53EZBPUSX5/action/replication_record"}},"created_at":"2026-05-18T01:26:04.313626+00:00","updated_at":"2026-05-18T01:26:04.313626+00:00"}