{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:MZT7U45N2UKYZAXLTBQIO5JAWD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"5482436632d255fc2361153606e65317f7f5aafeeb4f68e97c27cad540af020e","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-09-26T22:38:20Z","title_canon_sha256":"34545ba44036218b79ae6103da59102749decfd4fbdbbcfa6691f5d7eefdd41e"},"schema_version":"1.0","source":{"id":"0809.4726","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0809.4726","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"arxiv_version","alias_value":"0809.4726v2","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0809.4726","created_at":"2026-05-18T04:41:16Z"},{"alias_kind":"pith_short_12","alias_value":"MZT7U45N2UKY","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_16","alias_value":"MZT7U45N2UKYZAXL","created_at":"2026-05-18T12:25:57Z"},{"alias_kind":"pith_short_8","alias_value":"MZT7U45N","created_at":"2026-05-18T12:25:57Z"}],"graph_snapshots":[{"event_id":"sha256:a4e0f96426bf2fc63ce70f65629d5bca838f0c059da7fd448bac0abaad4bbc40","target":"graph","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider the $t$-improper chromatic number of the Erd{\\H o}s-R{\\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.","authors_text":"Colin McDiarmid, Ross J. Kang","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-09-26T22:38:20Z","title":"The t-improper chromatic number of random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0809.4726","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6a339dbdc6bad79ce37c699f8820f012e825613659f0ba8d17fa654252e0b1f7","target":"record","created_at":"2026-05-18T04:41:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5482436632d255fc2361153606e65317f7f5aafeeb4f68e97c27cad540af020e","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2008-09-26T22:38:20Z","title_canon_sha256":"34545ba44036218b79ae6103da59102749decfd4fbdbbcfa6691f5d7eefdd41e"},"schema_version":"1.0","source":{"id":"0809.4726","kind":"arxiv","version":2}},"canonical_sha256":"6667fa73add5158c82eb9860877520b0e45bd6e59f80e6484dd90f5a9d79e8b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6667fa73add5158c82eb9860877520b0e45bd6e59f80e6484dd90f5a9d79e8b5","first_computed_at":"2026-05-18T04:41:16.482673Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:16.482673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hhVExRXOdvUjqVKGvNyAo6C2j6zF9DXKVDvCD1Q4/ehOnC/NgHNB792qo4rebkBYBE4D12ShOLwT/WSkK/hEDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:16.483276Z","signed_message":"canonical_sha256_bytes"},"source_id":"0809.4726","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6a339dbdc6bad79ce37c699f8820f012e825613659f0ba8d17fa654252e0b1f7","sha256:a4e0f96426bf2fc63ce70f65629d5bca838f0c059da7fd448bac0abaad4bbc40"],"state_sha256":"096c03cc9bd51ea4f4ebd054ff4e3b80e147c67ee0db697b626e1a5af0b3892c"}