{"paper":{"title":"The genus of the Erd\\H{o}s-R\\'enyi random graph and the fragile genus property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chris Dowden, Michael Krivelevich, Mihyun Kang","submitted_at":"2018-06-14T11:20:24Z","abstract_excerpt":"We investigate the genus $g(n,m)$ of the Erd\\H{o}s-R\\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$ falls into.\n  Results already exist for $m \\le \\frac{n}{2} + O(n^{2/3})$ and $m = \\omega \\left( n^{1+\\frac{1}{j}} \\right)$ for $j \\in \\mathbb{N}$, and so we focus on the intermediate cases. We establish that $g(n,m) = (1+o(1)) \\frac{m}{2}$ whp (with high probability) when $n \\ll m = n^{1+o(1)}$, that $g(n,m) = (1+o(1)) \\mu (\\lambda) m$ whp for a given functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.05468","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"}