{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:OQSGKZMZB227BOJ7GTDIY3IYBX","short_pith_number":"pith:OQSGKZMZ","schema_version":"1.0","canonical_sha256":"74246565990eb5f0b93f34c68c6d180dfff99cca4f8fea8b20722a6bf59d93ee","source":{"kind":"arxiv","id":"1509.08251","version":1},"attestation_state":"computed","paper":{"title":"Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Christoph Berkholz, Martin Grohe, Paul Bonsma","submitted_at":"2015-09-28T09:36:08Z","abstract_excerpt":"An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of colours. This is a widely used subroutine for graph isomorphism testing algorithms, since any automorphism needs to be colour preserving. We give an $O((m+n)\\log n)$ algorithm for finding a canonical version of such a stable colouring, on graphs with $n$ vertices and $m$ edges. We show that no faster algorithm is possible, under some modest assumptions about the ty"},"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":"1509.08251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2015-09-28T09:36:08Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"9fedd9de13e90ab5b82b92f6ec4093d25eadd676418a892e4e4d1bd8e8255a9c","abstract_canon_sha256":"0a3ef8ec108f42f838cc3128e58401b1502bec38563952685271deffa002e3e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:54.310442Z","signature_b64":"S31ihACMpQRE+uXDDwtqP1iEu6H8dHA6ToqKlPfFVe13qHGlkOq70kiLbooqQiG21jXNNes/n1aIHyqgcmeJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74246565990eb5f0b93f34c68c6d180dfff99cca4f8fea8b20722a6bf59d93ee","last_reissued_at":"2026-05-18T01:31:54.309850Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:54.309850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.DS","authors_text":"Christoph Berkholz, Martin Grohe, Paul Bonsma","submitted_at":"2015-09-28T09:36:08Z","abstract_excerpt":"An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of colours. This is a widely used subroutine for graph isomorphism testing algorithms, since any automorphism needs to be colour preserving. We give an $O((m+n)\\log n)$ algorithm for finding a canonical version of such a stable colouring, on graphs with $n$ vertices and $m$ edges. We show that no faster algorithm is possible, under some modest assumptions about the ty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08251","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":"1509.08251","created_at":"2026-05-18T01:31:54.309931+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.08251v1","created_at":"2026-05-18T01:31:54.309931+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.08251","created_at":"2026-05-18T01:31:54.309931+00:00"},{"alias_kind":"pith_short_12","alias_value":"OQSGKZMZB227","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"OQSGKZMZB227BOJ7","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"OQSGKZMZ","created_at":"2026-05-18T12:29:34.919912+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/OQSGKZMZB227BOJ7GTDIY3IYBX","json":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX.json","graph_json":"https://pith.science/api/pith-number/OQSGKZMZB227BOJ7GTDIY3IYBX/graph.json","events_json":"https://pith.science/api/pith-number/OQSGKZMZB227BOJ7GTDIY3IYBX/events.json","paper":"https://pith.science/paper/OQSGKZMZ"},"agent_actions":{"view_html":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX","download_json":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX.json","view_paper":"https://pith.science/paper/OQSGKZMZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.08251&json=true","fetch_graph":"https://pith.science/api/pith-number/OQSGKZMZB227BOJ7GTDIY3IYBX/graph.json","fetch_events":"https://pith.science/api/pith-number/OQSGKZMZB227BOJ7GTDIY3IYBX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX/action/storage_attestation","attest_author":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX/action/author_attestation","sign_citation":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX/action/citation_signature","submit_replication":"https://pith.science/pith/OQSGKZMZB227BOJ7GTDIY3IYBX/action/replication_record"}},"created_at":"2026-05-18T01:31:54.309931+00:00","updated_at":"2026-05-18T01:31:54.309931+00:00"}