{"paper":{"title":"Asymptotics of the Upper Matching Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeff Kahn, Liviu Ilinca","submitted_at":"2012-05-19T16:03:05Z","abstract_excerpt":"We give upper bounds for the number $\\Phi_\\ell(G)$ of matchings of size $\\ell$ in (i) bipartite graphs $G=(X\\cup Y, E)$ with specified degrees $d_x$ ($x\\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular, for $d$-regular, $N$-vertex graphs, our bound is best possible up to an error factor of the form $\\exp[o_d(1)N]$, where $o_d(1) \\rightarrow 0$ as $d \\rightarrow \\infty$. This represents the best progress to date on the \"Upper Matching Conjecture\" of Friedland, Krop, Lundow and Markstr\\\"om. Some further possibilities are also suggested."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4342","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"}