The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs
Niranjan Balachandran, Deepanshu Kush
备注：
28 pages, 3 figures; Changes from v1 to v2: improved exposition and clarity of proofs, added references. Change from v2 to v3: added one new reference for Euclidean trees
发表信息：
The Electronic Journal of Combinatorics, Volume 28, Issue 2, Article P2.48 (2021)
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 Erdős-Rényi 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.
Let $X=(V, E)$ be a finite regular graph and $H_t(u, v), \, u, v \in V$, the heat kernel on $X$. We prove that, if the graph $X$ is bipartite and has four distinct Laplacian eigenvalues, the ratio $H_t(u, v)/H_t(u, u), \, u, v \in V,$ is monotonically non-decreasing as a function of $t$. The key to the proof is the fact that such a graph is an incidence graph of a symmetric 2-design.
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Eindhoven University of Technology（埃因霍温理工大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Hamburg University of Technology（汉堡工业大学）；Universidad de Valladolid（巴利亚多利德大学）
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