{"paper":{"title":"The evolution of random graphs on surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chris Dowden, Mihyun Kang, Philipp Spr\\\"ussel","submitted_at":"2017-09-04T08:52:07Z","abstract_excerpt":"For integers $g,m \\geq 0$ and $n>0$, let $S_{g}(n,m)$ denote the graph taken uniformly at random from the set of all graphs on $\\{1,2, \\ldots, n\\}$ with exactly $m=m(n)$ edges and with genus at most $g$. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of $S_{g}(n,m)$, finding that there is often different asymptotic behaviour depending on the ratio $\\frac{m}{n}$.\n  In our main results, we show that the probability that $S_{g}(n,m)$ contains any given non-planar component converges to $0$ as $n \\to \\infty$ for all $m(n)$; the probability"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00864","kind":"arxiv","version":2},"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"}