{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:B2DVUXSIQ4HETPNKKGWHROQCGH","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":"12e7e36313ca6f53259d456017b8e2e8b00b250552b310113ccafd6e4fc257d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-04T10:10:35Z","title_canon_sha256":"b6a423d7f0a6bebd2b7d4aa07196f6ed6009617f2e05b6f64a9fa435ff655bfb"},"schema_version":"1.0","source":{"id":"1108.1036","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.1036","created_at":"2026-05-18T04:16:13Z"},{"alias_kind":"arxiv_version","alias_value":"1108.1036v1","created_at":"2026-05-18T04:16:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.1036","created_at":"2026-05-18T04:16:13Z"},{"alias_kind":"pith_short_12","alias_value":"B2DVUXSIQ4HE","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"B2DVUXSIQ4HETPNK","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"B2DVUXSI","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:a7fa236862e9966a36b401aace8320d5b16c287f37baae622667f47ca0dfebbb","target":"graph","created_at":"2026-05-18T04:16:13Z","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":"The colouring number col(G) of a graph G is the smallest integer k for which there is an ordering of the vertices of G such that when removing the vertices of G in the specified order no vertex of degree more than k-1 in the remaining graph is removed at any step. An edge e of a graph G is said to be double-col-critical if the colouring number of G-V(e) is at most the colouring number of G minus 2. A connected graph G is said to be double-col-critical if each edge of G is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove ","authors_text":"Anders Sune Pedersen, Matthias Kriesell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-04T10:10:35Z","title":"On graphs double-critical with respect to the colouring number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.1036","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:ac3df4e0fb6f42e1d31e6430c284cd4c739f01deda290651610b91e795dec9a2","target":"record","created_at":"2026-05-18T04:16:13Z","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":"12e7e36313ca6f53259d456017b8e2e8b00b250552b310113ccafd6e4fc257d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-04T10:10:35Z","title_canon_sha256":"b6a423d7f0a6bebd2b7d4aa07196f6ed6009617f2e05b6f64a9fa435ff655bfb"},"schema_version":"1.0","source":{"id":"1108.1036","kind":"arxiv","version":1}},"canonical_sha256":"0e875a5e48870e49bdaa51ac78ba0231ee2a7324974f841154fc268d7118a260","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e875a5e48870e49bdaa51ac78ba0231ee2a7324974f841154fc268d7118a260","first_computed_at":"2026-05-18T04:16:13.187773Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:13.187773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dif0junQXYxzdx21YPPBRXWzFzv0HKnr1slSHJZzYREdOVQyLxkgh+n0bmJdkiLKpxl4+g5UopMdLI9KXWuEBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:13.188537Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.1036","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ac3df4e0fb6f42e1d31e6430c284cd4c739f01deda290651610b91e795dec9a2","sha256:a7fa236862e9966a36b401aace8320d5b16c287f37baae622667f47ca0dfebbb"],"state_sha256":"f0142840b06b0a93bcc5cf6a6e0ad07aa1179f7a4156e8758667074bd6c7747b"}