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In this paper we determine $n(\\Sigma)$, the order of a minimal quadrangulation of a surface $\\Sigma$, for all surfaces, both orientable and nonorientable. Letting $S_0$ denote the sphere and $N_2$ the Klein bottle, we prove that $n(S_0)=4, n(N_2)=6$, and $n(\\Sigma)=\\lceil (5+\\sqrt{25-16\\chi(\\Sig"},"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":"2106.13377","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-06-25T01:10:43Z","cross_cats_sorted":[],"title_canon_sha256":"2445272ab4d3e568feea82bc9f0eb75a2555fb0a33e9cc120e5c0273815da906","abstract_canon_sha256":"7fc5b39d326e80d38c63452c0c63487c367e337c6cc0ecb040d3cf03e6404c0d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:52:18.827417Z","signature_b64":"ELT9Dk7CpYNMNswaau9eD9XOlpucBu/afRVIXMIMBdjCZPZTMoTTmWe1u/UF/Yqlw6wakZrGKKhxiwrwQCDXBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6da4294b776f9098267ab7fb1a6f55cae9c0386a5bd2ae496bd074e3342aaad","last_reissued_at":"2026-07-05T02:52:18.827004Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:52:18.827004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal quadrangulations of surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dong Ye, M. 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