{"paper":{"title":"Phase transitions in graphs on orientable surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Mo{\\ss}hammer, Mihyun Kang, Philipp Spr\\\"ussel","submitted_at":"2017-08-25T09:54:00Z","abstract_excerpt":"Let $\\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\\{1,\\dotsc,n\\}$ with $m$ edges embeddable on $\\mathbb{S}_g$ features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd\\H{o}s--R\\'enyi random graph $G(n,m)$ chosen uniformly at random from all graphs with vertex set $[n]$ and $m$ edges. It takes place at $m=\\frac{n}{2}+O(n^{2/3})$, when a unique largest component, the so-called \\emph{giant compo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07671","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"}