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For a nonnegative integer $m$, let $f^{(m)}$ be the extension of $f$ to the graph $ G \\diamondplus \\overline{K_m}$ for which $f^{(m)}(v)=|V(G)|$ for each vertex $v$ of $\\overline{K_m}$. Let $m_c(G,f)$ be the minimum $m$ such that $ G \\diamondplus \\overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \\diamondplus \\overline{K_m}$ is not $f^{(m)}"},"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":"1711.02852","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-08T07:31:02Z","cross_cats_sorted":[],"title_canon_sha256":"68d99134c36a510792b5b76c611d6771e57ffe9d27e9d2121ef35ad7ef4d4e7c","abstract_canon_sha256":"b2aa98b1a1605d23780fecfdc0c113c8ee26b07dbee6da594db484ef64c01fed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:01.524212Z","signature_b64":"VWC7rlfvgjya01EWN/JtDWp+jpVWES+EcbIsaN15+I4Qpx7C5lmdyoxNTarVzqdE2/ja+juU6R2QQOZ+HxCgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"148b00d857c8957d30687788ecb01a83ccc45ebf2eacd09fc97d3564e2b2d650","last_reissued_at":"2026-05-18T00:31:01.523723Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:01.523723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"List colouring of graphs and generalized Dyck paths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rongxing Xu, Xuding Zhu, Yeong-Nan Yeh","submitted_at":"2017-11-08T07:31:02Z","abstract_excerpt":"The Catalan numbers occur in various counting problems in combinatorics. 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Let $m_c(G,f)$ be the minimum $m$ such that $ G \\diamondplus \\overline{K_m}$ is not $f^{(m)}$-choosable and $m_p(G,f)$ be the minimum $m$ such that $ G \\diamondplus \\overline{K_m}$ is not $f^{(m)}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02852","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":"1711.02852","created_at":"2026-05-18T00:31:01.523788+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.02852v1","created_at":"2026-05-18T00:31:01.523788+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.02852","created_at":"2026-05-18T00:31:01.523788+00:00"},{"alias_kind":"pith_short_12","alias_value":"CSFQBWCXZCKX","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CSFQBWCXZCKX2MDI","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CSFQBWCX","created_at":"2026-05-18T12:31:10.602751+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/CSFQBWCXZCKX2MDIO6EOZMA2QP","json":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP.json","graph_json":"https://pith.science/api/pith-number/CSFQBWCXZCKX2MDIO6EOZMA2QP/graph.json","events_json":"https://pith.science/api/pith-number/CSFQBWCXZCKX2MDIO6EOZMA2QP/events.json","paper":"https://pith.science/paper/CSFQBWCX"},"agent_actions":{"view_html":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP","download_json":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP.json","view_paper":"https://pith.science/paper/CSFQBWCX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.02852&json=true","fetch_graph":"https://pith.science/api/pith-number/CSFQBWCXZCKX2MDIO6EOZMA2QP/graph.json","fetch_events":"https://pith.science/api/pith-number/CSFQBWCXZCKX2MDIO6EOZMA2QP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP/action/storage_attestation","attest_author":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP/action/author_attestation","sign_citation":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP/action/citation_signature","submit_replication":"https://pith.science/pith/CSFQBWCXZCKX2MDIO6EOZMA2QP/action/replication_record"}},"created_at":"2026-05-18T00:31:01.523788+00:00","updated_at":"2026-05-18T00:31:01.523788+00:00"}