{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:CTZ6ZAG2TYH7QHLKYS7EFFP6ME","short_pith_number":"pith:CTZ6ZAG2","canonical_record":{"source":{"id":"1306.0140","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T19:20:59Z","cross_cats_sorted":[],"title_canon_sha256":"00b4dfe6a842228b7612a4756fa987fefc1a95667c5156016eff3ee63e039472","abstract_canon_sha256":"3982667abb3768a8043fd0cebe360f46cfa3773f3f7817c76a2c98b5ec466f81"},"schema_version":"1.0"},"canonical_sha256":"14f3ec80da9e0ff81d6ac4be4295fe6132e3f8ed883c854514ad66c4e70816ab","source":{"kind":"arxiv","id":"1306.0140","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0140","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0140v1","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0140","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"pith_short_12","alias_value":"CTZ6ZAG2TYH7","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CTZ6ZAG2TYH7QHLK","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CTZ6ZAG2","created_at":"2026-05-18T12:27:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:CTZ6ZAG2TYH7QHLKYS7EFFP6ME","target":"record","payload":{"canonical_record":{"source":{"id":"1306.0140","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T19:20:59Z","cross_cats_sorted":[],"title_canon_sha256":"00b4dfe6a842228b7612a4756fa987fefc1a95667c5156016eff3ee63e039472","abstract_canon_sha256":"3982667abb3768a8043fd0cebe360f46cfa3773f3f7817c76a2c98b5ec466f81"},"schema_version":"1.0"},"canonical_sha256":"14f3ec80da9e0ff81d6ac4be4295fe6132e3f8ed883c854514ad66c4e70816ab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:56.031910Z","signature_b64":"JvOXuTPgADVkkQyneJEW5nsSphKaQsDwML+QmnOYmZAyo14QZGpruUTNzL1kbr6UgZcqcOVREiWoO9Q1+CACBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14f3ec80da9e0ff81d6ac4be4295fe6132e3f8ed883c854514ad66c4e70816ab","last_reissued_at":"2026-05-18T03:21:56.031516Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:56.031516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.0140","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5YPGsAaLA5xHFkExsNBrtBAi1YPsGnGBzqNWTi+4P9Rq6NFN4bXU/OrXDrjMCGvvnmxcc0Ibhe9TTcnGpNTCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:08:07.944036Z"},"content_sha256":"face64a560e76acb5ec89cb581266e2f7a0d2257118abd6dc3be4eefd1beeeaf","schema_version":"1.0","event_id":"sha256:face64a560e76acb5ec89cb581266e2f7a0d2257118abd6dc3be4eefd1beeeaf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:CTZ6ZAG2TYH7QHLKYS7EFFP6ME","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nested colourings of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Cook II","submitted_at":"2013-06-01T19:20:59Z","abstract_excerpt":"A proper vertex colouring of a graph is \\emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time.\n  Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0140","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j4ZWBi1229olaT7mny2n8UxTMHZm3mMKORm1xQhoQbrDWpn4+/FKjD/j1roqQSa/Wl8Ifknu4p57wO3NJSeZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T22:08:07.944723Z"},"content_sha256":"cb36d0de211cc610e0448110bf7a6df28382927ddcc32ddbdb402e0b00bfc162","schema_version":"1.0","event_id":"sha256:cb36d0de211cc610e0448110bf7a6df28382927ddcc32ddbdb402e0b00bfc162"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/bundle.json","state_url":"https://pith.science/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T22:08:07Z","links":{"resolver":"https://pith.science/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME","bundle":"https://pith.science/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/bundle.json","state":"https://pith.science/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CTZ6ZAG2TYH7QHLKYS7EFFP6ME/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:CTZ6ZAG2TYH7QHLKYS7EFFP6ME","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":"3982667abb3768a8043fd0cebe360f46cfa3773f3f7817c76a2c98b5ec466f81","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T19:20:59Z","title_canon_sha256":"00b4dfe6a842228b7612a4756fa987fefc1a95667c5156016eff3ee63e039472"},"schema_version":"1.0","source":{"id":"1306.0140","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.0140","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"arxiv_version","alias_value":"1306.0140v1","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0140","created_at":"2026-05-18T03:21:56Z"},{"alias_kind":"pith_short_12","alias_value":"CTZ6ZAG2TYH7","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_16","alias_value":"CTZ6ZAG2TYH7QHLK","created_at":"2026-05-18T12:27:40Z"},{"alias_kind":"pith_short_8","alias_value":"CTZ6ZAG2","created_at":"2026-05-18T12:27:40Z"}],"graph_snapshots":[{"event_id":"sha256:cb36d0de211cc610e0448110bf7a6df28382927ddcc32ddbdb402e0b00bfc162","target":"graph","created_at":"2026-05-18T03:21:56Z","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":"A proper vertex colouring of a graph is \\emph{nested} if the vertices of each of its colour classes can be ordered by inclusion of their open neighbourhoods. Through a relation to partially ordered sets, we show that the nested chromatic number can be computed in polynomial time.\n  Clearly, the nested chromatic number is an upper bound for the chromatic number of a graph. We develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behaviour of the nested chromatic number under several graph operations, including the direct, Cartes","authors_text":"David Cook II","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T19:20:59Z","title":"Nested colourings of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0140","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:face64a560e76acb5ec89cb581266e2f7a0d2257118abd6dc3be4eefd1beeeaf","target":"record","created_at":"2026-05-18T03:21:56Z","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":"3982667abb3768a8043fd0cebe360f46cfa3773f3f7817c76a2c98b5ec466f81","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-01T19:20:59Z","title_canon_sha256":"00b4dfe6a842228b7612a4756fa987fefc1a95667c5156016eff3ee63e039472"},"schema_version":"1.0","source":{"id":"1306.0140","kind":"arxiv","version":1}},"canonical_sha256":"14f3ec80da9e0ff81d6ac4be4295fe6132e3f8ed883c854514ad66c4e70816ab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14f3ec80da9e0ff81d6ac4be4295fe6132e3f8ed883c854514ad66c4e70816ab","first_computed_at":"2026-05-18T03:21:56.031516Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:56.031516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JvOXuTPgADVkkQyneJEW5nsSphKaQsDwML+QmnOYmZAyo14QZGpruUTNzL1kbr6UgZcqcOVREiWoO9Q1+CACBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:56.031910Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.0140","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:face64a560e76acb5ec89cb581266e2f7a0d2257118abd6dc3be4eefd1beeeaf","sha256:cb36d0de211cc610e0448110bf7a6df28382927ddcc32ddbdb402e0b00bfc162"],"state_sha256":"9b424cfdd86a0626f9adee97194d8f8fd89ef18aa2b89383eb5033d052537ecf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IRzc1lMxbYGuOJzg9uRPYUZo749g9ALjMvoCIJl7+SMCwfhXor5cFFyp/AVkUqFQ7iT1T0tmNDC86ONqTFZmAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T22:08:07.948127Z","bundle_sha256":"dfead00534ff8ffee6a1ad92bff6671a9ddbe045cc8cfdc48957878ec883bd1f"}}