scholarly journals Normalized matching property of a class of posets consisting of subspaces

2007 ◽  
Vol 2 ◽  
pp. 309-318
Author(s):  
Huajun Zhang
Keyword(s):  
Normalized Matching Property

scholarly journals A Finite Word Poset

10.37236/1607 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Péter L. Erdős ◽  
Péter Sziklai ◽  
David C. Torney
Keyword(s):  
Automorphism Groups ◽  
Partial Ordering ◽  
Finite Alphabet ◽  
Combinatorial Properties ◽  
Normalized Matching Property ◽  
Finite Word

Our word posets have finite words of bounded length as their elements, with the words composed from a finite alphabet. Their partial ordering follows from the inclusion of a word as a subsequence of another word. The elemental combinatorial properties of such posets are established. Their automorphism groups are determined (along with similar result for the word poset studied by Burosch, Frank and Röhl [4]) and a BLYM inequality is verified (via the normalized matching property).

The normalized matching property from the generalized Macaulay theorem

Order ◽  
1995 ◽  
Vol 12 (3) ◽  
pp. 233-237
Author(s):  
G. F. Clements
Keyword(s):  
Normalized Matching Property

scholarly journals The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs

10.37236/9148 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Niranjan Balachandran ◽  
Deepanshu Kush
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Decomposition Theorem ◽  
Discrete Math ◽  
Bipartite Graphs ◽  
Classic Result ◽  
Normalized Matching Property ◽  
Vertex Partition ◽  
Vertex Set

A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matching Property (NMP) if for any subset $S\subseteq X$ we have $\frac{|N(S)|}{|Y|}\geq\frac{|S|}{|X|}$. In this paper, we prove the following results about the Normalized Matching Property.  The random bipartite graph $\mathbb{G}(k,n,p)$ with $|X|=k,|Y|=n$, and $k\leq n<\exp(k)$, and each pair $(x,y)\in X\times Y$ being an edge in $\mathbb{G}$ independently with probability $p$ has $p=\frac{\log n}{k}$ as the threshold for NMP. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for the existence of a perfect matching in $\mathbb{G}(n,n,p)$. 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 every $x\in X$ has degree at least $pn$ and every pair of distinct $x, x'\in X$ have at most $(1+\varepsilon)p^2n$ common neighbours. We show that Thomason pseudorandom graphs have the following property: Given $\varepsilon>0$ and $n\geq k\gg 0$, there exist functions $f,g$ with $f(x), g(x)\to 0$ as $x\to 0$, and sets $\mathrm{Del}_X\subset X, \  \mathrm{Del}_Y\subset Y$ with $|\mathrm{Del}_X|\leq f(\varepsilon)k,\ |\mathrm{Del}_Y|\leq g(\varepsilon)n$ such that $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP. Enroute, we prove an 'almost' vertex decomposition theorem: Every Thomason pseudorandom bipartite graph $G(X,Y)$ admits - except for a negligible portion of its vertex set - a partition of its vertex set into graphs that are spanned by trees that have NMP, and which arise organically through the Euclidean GCD algorithm. 

Sign in / Sign up

Export Citation Format

Share Document