{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:S6PGR3IFJQ27WFOZQW7CA6SDGQ","short_pith_number":"pith:S6PGR3IF","schema_version":"1.0","canonical_sha256":"979e68ed054c35fb15d985be207a43340df23dad9dfd31946d4b1f6e7f8d3077","source":{"kind":"arxiv","id":"2108.00001","version":1},"attestation_state":"computed","paper":{"title":"Mixing colourings in $2K_2$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Carl Feghali, Owen Merkel","submitted_at":"2021-08-02T15:04:31Z","abstract_excerpt":"The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any $k$-colourable $P_4$-free graph $G$, Bonamy and Bousquet proved that $R_{k+1}(G)$ is connected. In this short note, we complete the classification of the connectedness of $R_{k+1}(G)$ for a $k$-colourable graph $G$ excluding a fixed path, by constructing a $7$-chromatic $2K_2$-free (and hence $P_5$-free) graph admitting a frozen $8$-colouring. This settles a"},"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":"2108.00001","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-02T15:04:31Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"f648a18333a741f6de1c1b319c17b4e03c847e625c1fcefc1ece2f46e048d57b","abstract_canon_sha256":"ab6300b5cac4a7ed9eb33a034e647eb98cad78c1eb1e05c241217f25edacb710"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:02:18.311391Z","signature_b64":"NQnOoCb5kWqYixAat6lOggfYCFjdEai8TjgY5xEd3ixn/XDI4j0tVvndTvesCo5L3JToPgeQilwmmoyR1f4LBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"979e68ed054c35fb15d985be207a43340df23dad9dfd31946d4b1f6e7f8d3077","last_reissued_at":"2026-07-05T03:02:18.310866Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:02:18.310866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mixing colourings in $2K_2$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Carl Feghali, Owen Merkel","submitted_at":"2021-08-02T15:04:31Z","abstract_excerpt":"The reconfiguration graph for the $k$-colourings of a graph $G$, denoted $R_{k}(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. For any $k$-colourable $P_4$-free graph $G$, Bonamy and Bousquet proved that $R_{k+1}(G)$ is connected. In this short note, we complete the classification of the connectedness of $R_{k+1}(G)$ for a $k$-colourable graph $G$ excluding a fixed path, by constructing a $7$-chromatic $2K_2$-free (and hence $P_5$-free) graph admitting a frozen $8$-colouring. This settles a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.00001","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.00001/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2108.00001","created_at":"2026-07-05T03:02:18.310951+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.00001v1","created_at":"2026-07-05T03:02:18.310951+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.00001","created_at":"2026-07-05T03:02:18.310951+00:00"},{"alias_kind":"pith_short_12","alias_value":"S6PGR3IFJQ27","created_at":"2026-07-05T03:02:18.310951+00:00"},{"alias_kind":"pith_short_16","alias_value":"S6PGR3IFJQ27WFOZ","created_at":"2026-07-05T03:02:18.310951+00:00"},{"alias_kind":"pith_short_8","alias_value":"S6PGR3IF","created_at":"2026-07-05T03:02:18.310951+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/S6PGR3IFJQ27WFOZQW7CA6SDGQ","json":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ.json","graph_json":"https://pith.science/api/pith-number/S6PGR3IFJQ27WFOZQW7CA6SDGQ/graph.json","events_json":"https://pith.science/api/pith-number/S6PGR3IFJQ27WFOZQW7CA6SDGQ/events.json","paper":"https://pith.science/paper/S6PGR3IF"},"agent_actions":{"view_html":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ","download_json":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ.json","view_paper":"https://pith.science/paper/S6PGR3IF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.00001&json=true","fetch_graph":"https://pith.science/api/pith-number/S6PGR3IFJQ27WFOZQW7CA6SDGQ/graph.json","fetch_events":"https://pith.science/api/pith-number/S6PGR3IFJQ27WFOZQW7CA6SDGQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ/action/storage_attestation","attest_author":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ/action/author_attestation","sign_citation":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ/action/citation_signature","submit_replication":"https://pith.science/pith/S6PGR3IFJQ27WFOZQW7CA6SDGQ/action/replication_record"}},"created_at":"2026-07-05T03:02:18.310951+00:00","updated_at":"2026-07-05T03:02:18.310951+00:00"}