scholarly journals Sharp Threshold Functions for Random Intersection Graphs via a Coupling Method

10.37236/523 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Perfect Matching ◽  
Hamilton Cycle ◽  
Coupling Method ◽  
New Method ◽  
Intersection Graphs ◽  
Sharp Threshold ◽  
Threshold Functions

We present a new method which enables us to find threshold functions for many properties in random intersection graphs. This method is used to establish sharp threshold functions in random intersection graphs for $k$–connectivity, perfect matching containment and Hamilton cycle containment.

scholarly journals The Coupling Method for Inhomogeneous Random Intersection Graphs.

10.37236/5186 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Katarzyna Rybarczyk
Keyword(s):  
Coupling Method ◽  
Intersection Graphs ◽  
New Approach ◽  
Threshold Functions ◽  
Wide Family

We present new results concerning threshold functions for a wide family of random intersection graphs. To this end we improve and generalize  the coupling method introduced for random intersection graphs so that it may be used for a wider range of parameters. Using the new approach we are able to tighten the best known results concerning random intersection graphs and establish threshold functions for some monotone properties of inhomogeneous random intersection graphs. Considered properties are: $k$-connectivity, matching containment and hamiltonicity.

scholarly journals Matchings, Cycle Bases, and the Maximum Genus of a Graph

10.37236/2479 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Michal Kotrbčík ◽  
Martin Škoviera
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Perfect Matching ◽  
Intersection Graph ◽  
Matching Number ◽  
Maximum Genus ◽  
Cycle Space ◽  
Intersection Graphs ◽  
Connected Graphs ◽  
Cycle Bases

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is paid to cycle bases consisting of fundamental cycles with respect to a given spanning tree. For $4$-edge-connected graphs, the intersection graph with respect to any spanning tree (and, in fact, with respect to any basis) has either a perfect matching or a matching missing exactly one vertex. We show that if a graph is not $4$-edge-connected, different spanning trees may lead to intersection graphs with different matching numbers. We also show that there exist $2$-edge connected graphs for which the set of values of matching numbers of their intersection graphs contains arbitrarily large gaps.

Random high-dimensional orders

1995 ◽  
Vol 27 (01) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order P k (n) on n points is defined by taking n points uniformly at random from [0,1] k . Previous work has concentrated on the case where k is constant: we consider the model where k increases with n. We pay particular attention to the height H k (n) of P k (n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in P k (n): if k – (t/log t!) log n tends to + ∞ then the probability that P k (n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of H k (n) for the entire range of k(n). We also consider the maximum degree of P k (n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k. We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of P k (n) for large k(n).

Meshless - The Finite Element of a Direct Coupling Method Mechanical Problems in the Application

2012 ◽  
Vol 548 ◽  
pp. 421-424
Author(s):  
Jin Ping Wang ◽  
Yu Jing Gao ◽  
De Hua Wang
Keyword(s):  
Finite Element ◽  
Shape Function ◽  
Direct Coupling ◽  
Coupling Method ◽  
New Method ◽  
Field Function ◽  
The Core ◽  
Definition Of ◽  
Short Time ◽  
The Brain

Coupling method is developed in recent years to solve numerical problems a new method, meshless - the finite element of a direct coupling method is based on the definition of the generalized unit of coupling of the new method . The core of this method is the use of each unit in the shape function to the assumption that the brain that the whole sub-domain to be seeking to solve the unknown field function. Coupling with other compared with the method is simple to calculate the advantages of a short time.

scholarly journals Matchings and Hamilton cycles in hypergraphs

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  
Bipartite Graph ◽  
Perfect Matching ◽  
Minimum Degree ◽  
Hamilton Cycle ◽  
Uniform Hypergraph ◽  
Hamilton Cycles ◽  
Uniform Hypergraphs ◽  
International Audience

International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.

Random high-dimensional orders

1995 ◽  
Vol 27 (1) ◽  
pp. 161-184 ◽  
Author(s):  
Béla Bollobás ◽  
Graham Brightwell
Keyword(s):  
Partial Order ◽  
Rapid Growth ◽  
Entire Range ◽  
Maximum Degree ◽  
Threshold Function ◽  
High Dimensional ◽  
Sharp Threshold ◽  
Threshold Functions

The random k-dimensional partial order Pk(n) on n points is defined by taking n points uniformly at random from [0,1]k. Previous work has concentrated on the case where k is constant: we consider the model where k increases with n.We pay particular attention to the height Hk(n) of Pk(n). We show that k = (t/log t!) log n is a sharp threshold function for the existence of a t-chain in Pk(n): if k – (t/log t!) log n tends to + ∞ then the probability that Pk(n) contains a t-chain tends to 0; whereas if the quantity tends to − ∞ then the probability tends to 1. We describe the behaviour of Hk(n) for the entire range of k(n).We also consider the maximum degree of Pk(n). We show that, for each fixed d ≧ 2, is a threshold function for the appearance of an element of degree d. Thus the maximum degree undergoes very rapid growth near this value of k.We make some remarks on the existence of threshold functions in general, and give some bounds on the dimension of Pk(n) for large k(n).

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