{"paper":{"title":"The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Deepanshu Kush, Niranjan Balachandran","submitted_at":"2019-08-07T13:34:26Z","abstract_excerpt":"A simple generalization of the Hall's condition in bipartite graphs, the Normalized Matching Property (NMP) in a graph $G(X,Y,E)$ with vertex partition $(X,Y)$ states that for any subset $S\\subseteq X$, we have $\\frac{|N(S)|}{|Y|}\\ge\\frac{|S|}{|X|}$. In this paper, we show the following results about having the Normalized Matching Property in random and pseudorandom graphs.\n  1. We establish $p=\\frac{\\log n}{k}$ as a sharp threshold for having NMP in $\\mathbb{G}(k,n,p)$, which is the graph with $|X|=k,|Y|=n$ (assuming $k\\le n\\leq \\exp(o(k))$), and in which each pair $(x,y)\\in X\\times Y$ is an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02628","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02628/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}