scholarly journals Constraining the Clustering Transition for Colorings of Sparse Random Graphs

10.37236/7040 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Michael Anastos ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Hamming Distance ◽  
Giant Component ◽  
Vertex Set ◽  
Almost All

Let $\Omega_q$ denote the set of proper $[q]$-colorings of the random graph $G_{n,m}, m=dn/2$ and let $H_q$ be the graph with vertex set $\Omega_q$ and an edge $\{\sigma,\tau\}$ where $\sigma,\tau$ are mappings $[n]\to[q]$ iff $h(\sigma,\tau)=1$. Here $h(\sigma,\tau)$ is the Hamming distance $|\{v\in [n]:\sigma(v)\neq\tau(v)\}|$. We show that w.h.p. $H_q$ contains a single giant component containing almost all colorings in $\Omega_q$ if $d$ is sufficiently large and $q\geq \frac{cd}{\log d}$ for a constant $c>3/2$.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

scholarly journals The Total Acquisition Number of Random Graphs

10.37236/5327 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
High Probability ◽  
Sharp Threshold ◽  
Positive Weight ◽  
Minimum Value ◽  
Almost All

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.

On colouring random graphs

1975 ◽  
Vol 77 (2) ◽  
pp. 313-324 ◽  
Author(s):  
G. R. Grimmett ◽  
C. J. H. McDiarmid
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Subgraph ◽  
Vertex Set ◽  
Image Position

AbstractLet ωn denote a random graph with vertex set {1, 2, …, n}, such that each edge is present with a prescribed probability p, independently of the presence or absence of any other edges. We show that the number of vertices in the largest complete subgraph of ωn is, with probability one,

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

scholarly journals Metric Dimension for Random Graphs

10.37236/2639 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Dieter Mitsche ◽  
Paweł Prałat
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Metric Dimension ◽  
Wide Range ◽  
Minimum Number ◽  
Vertex Set

The metric dimension of a graph $G$ is the minimum number of vertices in a subset $S$ of the vertex set of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $S$. In this paper we investigate the metric dimension of the random graph $G(n,p)$ for a wide range of probabilities $p=p(n)$.

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (02) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

scholarly journals A Note on the Non-Colorability Threshold of a Random Graph

10.37236/1507 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Alexis C. Kaporis ◽  
Lefteris M. Kirousis ◽  
Yannis C. Stamatiou
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Sharp Estimates ◽  
A Value ◽  
Occupancy Problem ◽  
Random Placement ◽  
Almost All ◽  
The Relationship

In this paper we consider the problem of establishing a value $r_0$ such that almost all random graphs with $n$ vertices and $rn$ edges, $r > r_0$, are asymptotically not 3-colorable. In our approach we combine the concept of rigid legal colorings introduced by Achlioptas and Molloy with the occupancy problem for random allocations of balls into bins. Using the sharp estimates obtained by Kamath et al. of the probability that no bin is empty after the random placement of the balls and exploiting the relationship between the placement of balls and the rigid legal colorings, we improve the value $r_0 = 2.522$ previously obtained by Achlioptas and Molloy to $r_0 = 2.495$.

On the connectedness of a random graph

1984 ◽  
Vol 96 (1) ◽  
pp. 151-166 ◽  
Author(s):  
G. R. Grimmett ◽  
M. Keane ◽  
J. M. Marstrand
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Higher Dimensions ◽  
Greatest Common Divisor ◽  
Vertex Set ◽  
Vertex Sets

AbstractLet p = (p(i): i ≥ 0) be a sequence of numbers satisfying 0 ≤ p(i) < 1 for i = 0,1,2,…, and let G be a random graph with vertex set ℤ = {…, — 1, 0, 1,…} and with edge set defined as follows: for each pair i, j of vertices, where i ≤ j, there is an edge joining i and j with probability p(j — i), independently of the presence or absence of all other edges. We explore the connectedness of G, showing that G is almost surely connected if and only if Σip(i) = ∞ and the (positive) greatest common divisor of the set {i ≥ 1: p(i) < 0} equals 1; if one of these two conditions fails to hold then G is almost surely disconnected. Corresponding results hold in higher dimensions, for random graphs defined on the vertex sets ℤd where d ≥ 2.

scholarly journals Generating stationary random graphs on ℤ with prescribed independent, identically distributed degrees

2006 ◽  
Vol 38 (2) ◽  
pp. 287-298 ◽  
Author(s):  
Maria Deijfen ◽  
Ronald Meester
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Random Graphs ◽  
Random Number ◽  
Random Variables ◽  
Nearest Neighbors ◽  
Expected Length ◽  
Vertex Set ◽  
Finitary Isomorphisms ◽  
Independent Identically Distributed

Let F be a probability distribution with support on the nonnegative integers. We describe two algorithms for generating a stationary random graph, with vertex set ℤ, in which the degrees of the vertices are independent, identically distributed random variables with distribution F. Focus is on an algorithm generating a graph in which, initially, a random number of ‘stubs’ with distribution F is attached to each vertex. Each stub is then randomly assigned a direction (left or right) and the edge configuration obtained by pairing stubs pointing to each other, first exhausting all possible connections between nearest neighbors, then linking second-nearest neighbors, and so on. Under the assumption that F has finite mean, it is shown that this algorithm leads to a well-defined configuration, but that the expected length of the shortest edge attached to a given vertex is infinite. It is also shown that any stationary algorithm for pairing stubs with random, independent directions causes the total length of the edges attached to a given vertex to have infinite mean. Connections to the problem of constructing finitary isomorphisms between Bernoulli shifts are discussed.

scholarly journals Random Graphs with Few Disjoint Cycles

2011 ◽  
Vol 20 (5) ◽  
pp. 763-775 ◽  
Author(s):  
VALENTAS KURAUSKAS ◽  
COLIN McDIARMID
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Random Graph ◽  
Random Graphs ◽  
Small Proportion ◽  
Clique Number ◽  
Disjoint Cycles ◽  
Vertex Set ◽  
Exponentially Small

The classical Erdős–Pósa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G−B has no cycles. We show that, amongst all such graphs on vertex set {1,. . .,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number.A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős–Pósa theorem with the extra ‘redundancy’ property that B–v is still a blocker for all but at most k vertices v ∈ B.

Sign in / Sign up

Export Citation Format

Share Document