scholarly journals The Asymptotic Induced Matching Number of Hypergraphs: Balanced Binary Strings

10.37236/9019 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Srinivasan Arunachalam ◽  
Péter Vrana ◽  
Jeroen Zuiddam
Keyword(s):  
Upper Bounds ◽  
Algebraic Complexity ◽  
Hamming Weight ◽  
Matching Number ◽  
Induced Matching ◽  
Theoretical Computer ◽  
Uniform Hypergraphs ◽  
Tensor Theory ◽  
Algebraic Complexity Theory ◽  
Order Extension

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith–Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science.Our proof relies on a new combinatorial inequality that may be of independent interest. This inequality concerns how many pairs of Boolean vectors of fixed Hamming weight can have their sum in a fixed subspace.

scholarly journals The size of 3-uniform hypergraphs with given matching number and codegree

2019 ◽  
Vol 342 (3) ◽  
pp. 760-767
Author(s):  
Xinmin Hou ◽  
Lei Yu ◽  
Jun Gao ◽  
Boyuan Liu
Keyword(s):  
Matching Number ◽  
Uniform Hypergraphs

scholarly journals Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

scholarly journals Regularity of bicyclic graphs and their powers

2019 ◽  
Vol 19 (03) ◽  
pp. 2050057 ◽  
Author(s):  
Yairon Cid-Ruiz ◽  
Sepehr Jafari ◽  
Navid Nemati ◽  
Beatrice Picone
Keyword(s):  
Base Case ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Base Graph ◽  
Bicyclic Graph ◽  
Bicyclic Graphs

Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.

scholarly journals The Bounds of the Edge Number in Generalized Hypertrees

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 2 ◽  
Author(s):  
Ke Zhang ◽  
Haixing Zhao ◽  
Zhonglin Ye ◽  
Yu Zhu ◽  
Liang Wei
Keyword(s):  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Number Of Components ◽  
Uniform Hypergraphs ◽  
Vertex Set

A hypergraph H = ( V , ε ) is a pair consisting of a vertex set V , and a set ε of subsets (the hyperedges of H ) of V . A hypergraph H is r -uniform if all the hyperedges of H have the same cardinality r . Let H be an r -uniform hypergraph, we generalize the concept of trees for r -uniform hypergraphs. We say that an r -uniform hypergraph H is a generalized hypertree ( G H T ) if H is disconnected after removing any hyperedge E , and the number of components of G H T − E is a fixed value k   ( 2 ≤ k ≤ r ) . We focus on the case that G H T − E has exactly two components. An edge-minimal G H T is a G H T whose edge set is minimal with respect to inclusion. After considering these definitions, we show that an r -uniform G H T on n vertices has at least 2 n / ( r + 1 ) edges and it has at most n − r + 1 edges if r ≥ 3   and   n ≥ 3 , and the lower and upper bounds on the edge number are sharp. We then discuss the case that G H T − E has exactly k   ( 2 ≤ k ≤ r − 1 ) components.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals Invariant and geometric aspects of algebraic complexity theory I

1991 ◽  
Vol 11 (5-6) ◽  
pp. 455-469
Author(s):  
Jacques Morgensern
Keyword(s):  
Complexity Theory ◽  
Algebraic Complexity ◽  
Algebraic Complexity Theory

scholarly journals Some Complete and Intermediate Polynomials in Algebraic Complexity Theory

2017 ◽  
Vol 62 (3) ◽  
pp. 622-652 ◽  
Author(s):  
Meena Mahajan ◽  
Nitin Saurabh
Keyword(s):  
Complexity Theory ◽  
Algebraic Complexity ◽  
Algebraic Complexity Theory

scholarly journals Sharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number

2011 ◽  
Vol 54 (11-12) ◽  
pp. 2869-2879 ◽  
Author(s):  
Shuchao Li ◽  
Qin Zhao
Keyword(s):  
Upper Bounds ◽  
Matching Number ◽  
Zagreb Indices ◽  
Bicyclic Graphs
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