{"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. N. Ellingham, Wenzhong Liu","submitted_at":"2021-06-25T01:10:43Z","abstract_excerpt":"A quadrangular embedding of a graph in a surface $\\Sigma$, also known as a quadrangulation of $\\Sigma$, is a cellular embedding in which every face is bounded by a $4$-cycle. A quadrangulation of $\\Sigma$ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in $\\Sigma$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.13377","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/2106.13377/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"}