{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:4WV6HEFG4JYVL4O5RZYFUAPH45","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":"db2cba72baf351e88a384830f2b3d11c20e3ec18ee0937e5923c64e11ab32a7c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2018-03-20T23:42:47Z","title_canon_sha256":"3ba34a18b90039efc754c34ae4b6e5e55a31990722fc4bd34ad3e82ee81fbf57"},"schema_version":"1.0","source":{"id":"1803.07694","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.07694","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"arxiv_version","alias_value":"1803.07694v1","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.07694","created_at":"2026-05-18T00:20:29Z"},{"alias_kind":"pith_short_12","alias_value":"4WV6HEFG4JYV","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_16","alias_value":"4WV6HEFG4JYVL4O5","created_at":"2026-05-18T12:32:05Z"},{"alias_kind":"pith_short_8","alias_value":"4WV6HEFG","created_at":"2026-05-18T12:32:05Z"}],"graph_snapshots":[{"event_id":"sha256:44a033f93cd23bdabbbbc56a7a3ab8a9b70e215c1897b95592d4a1aa9f370e3b","target":"graph","created_at":"2026-05-18T00:20:29Z","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":"Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has \"defect\" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has \"clustering\" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerp","authors_text":"David R. Wood","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2018-03-20T23:42:47Z","title":"Defective and Clustered Graph Colouring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07694","kind":"arxiv","version":1},"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:edf5d45627b0269436e0c750428ebbd2dc0e50e04f8e7e3fe5a58f88f60547cf","target":"record","created_at":"2026-05-18T00:20:29Z","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":"db2cba72baf351e88a384830f2b3d11c20e3ec18ee0937e5923c64e11ab32a7c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2018-03-20T23:42:47Z","title_canon_sha256":"3ba34a18b90039efc754c34ae4b6e5e55a31990722fc4bd34ad3e82ee81fbf57"},"schema_version":"1.0","source":{"id":"1803.07694","kind":"arxiv","version":1}},"canonical_sha256":"e5abe390a6e27155f1dd8e705a01e7e77aae8b1f92e394ac9c03a58ded00c225","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e5abe390a6e27155f1dd8e705a01e7e77aae8b1f92e394ac9c03a58ded00c225","first_computed_at":"2026-05-18T00:20:29.247617Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:29.247617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cV9eHkMj6m1XZBszCoUzB4p7QiVg6cUgE8v9/pIDk7yuNxiQhP/y12r2D2xxTEDZ98lBEqEFsiWadt6jX0bLCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:29.248048Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.07694","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:edf5d45627b0269436e0c750428ebbd2dc0e50e04f8e7e3fe5a58f88f60547cf","sha256:44a033f93cd23bdabbbbc56a7a3ab8a9b70e215c1897b95592d4a1aa9f370e3b"],"state_sha256":"e9599e84bb18197e1da7326cee398711c3fa7a925ff446eb7b7a8735e7ea543c"}