# The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs

## TL;DR

This paper investigates the conditions under which bipartite graphs exhibit the Normalized Matching Property (NMP), establishing thresholds in random graphs and showing that pseudorandom graphs can be made to have NMP after minor vertex removals.

## Contribution

It introduces sharp thresholds for NMP in random bipartite graphs and demonstrates that pseudorandom graphs can attain NMP after removing a small fraction of vertices, extending understanding of graph matchings.

## Key findings

- Identifies p=log(n)/k as a sharp threshold for NMP in G(k,n,p).
- Shows pseudorandom bipartite graphs can have NMP after minor vertex deletions.
- Proves an almost vertex decomposition into Euclidean trees with NMP.

## Abstract

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. 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 edge independently with probability $p$. This generalizes a classic result of Erd\H{o}s-R\'enyi on the $\frac{\log n}{n}$ threshold for having a perfect matching in $\mathbb{G}(n,n,p)$. 2. We also show that a pseudorandom bipartite graph - upon deletion of a vanishingly small fraction of vertices - admits NMP, provided it is not too sparse. More precisely, a bipartite graph $G(X,Y)$, with $k=|X|\le |Y|=n$, is said to be Thomason pseudorandom (following A. Thomason (Discrete Math., 1989)) with parameters $(p,\varepsilon)$ if each $x\in X$ has degree at least $pn$ and each pair of distinct $x, x'\in X$ has at most $(1+\varepsilon)p^2n$ common neighbors. We show that for any large enough $(p,\varepsilon)$-Thomason pseudorandom graph $G(X,Y)$, there are "tiny" subsets $\mathrm{Del}_X\subset X, \ \mathrm{Del}_Y\subset Y$ such that the subgraph $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP, provided $p \gg\tfrac{1}{k}$. En route, we prove an "almost" vertex decomposition theorem: Every such Thomason pseudorandom graph admits - excluding a negligible portion of its vertex set - a partition of its vertex set into graphs that we call Euclidean trees. These are trees that have NMP, and which arise organically through the Euclidean GCD algorithm.