{"paper":{"title":"Asymptotic Enumeration of Graph Classes with Many Components","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Panagiotou, Leon Ramzews","submitted_at":"2018-01-14T14:12:24Z","abstract_excerpt":"We consider graph classes $\\mathcal G$ in which every graph has components in a class $\\mathcal{C}$ of connected graphs. We provide a framework for the asymptotic study of $\\lvert\\mathcal{G}_{n,N}\\rvert$, the number of graphs in $\\mathcal{G}$ with $n$ vertices and $N:=\\lfloor\\lambda n\\rfloor$ components, where $\\lambda\\in(0,1)$. Assuming that the number of graphs with $n$ vertices in $\\mathcal{C}$ satisfies \\begin{align*} \\lvert \\mathcal{C}_n\\rvert\\sim b n^{-(1+\\alpha)}\\rho^{-n}n!, \\quad n\\to \\infty \\end{align*} for some $b,\\rho>0$ and $\\alpha>1$ -- a property commonly encountered in graph enu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04559","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"}