scholarly journals Coloring the Edges of a Random Graph without a Monochromatic Giant Component

10.37236/405 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Reto Spöhel ◽  
Angelika Steger ◽  
Henning Thomas
Keyword(s):  
Random Graph ◽  
Giant Component ◽  
Random Partition ◽  
Sharp Threshold ◽  
Color Class

We study the following two problems: i) Given a random graph $G_{n, m}$ (a graph drawn uniformly at random from all graphs on $n$ vertices with exactly $m$ edges), can we color its edges with $r$ colors such that no color class contains a component of size $\Theta(n)$? ii) Given a random graph $G_{n,m}$ with a random partition of its edge set into sets of size $r$, can we color its edges with $r$ colors subject to the restriction that every color is used for exactly one edge in every set of the partition such that no color class contains a component of size $\Theta(n)$? We prove that for any fixed $r\geq 2$, in both settings the (sharp) threshold for the existence of such a coloring coincides with the known threshold for $r$-orientability of $G_{n,m}$, which is at $m=rc_r^*n$ for some analytically computable constant $c_r^*$. The fact that the two problems have the same threshold is in contrast with known results for the two corresponding Achlioptas-type problems.

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.

The cover time of the giant component of a random graph

2008 ◽  
Vol 32 (4) ◽  
pp. 401-439 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Giant Component ◽  
Cover Time

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.

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 Convergence of Achlioptas Processes via Differential Equations with Unique Solutions

2015 ◽  
Vol 25 (1) ◽  
pp. 154-171 ◽  
Author(s):  
OLIVER RIORDAN ◽  
LUTZ WARNKE
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Large Class ◽  
Unique Solution ◽  
Random Graph ◽  
Giant Component ◽  
General Idea ◽  
Product Rule ◽  
System Of Differential Equations ◽  
Empty Graph

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

scholarly journals The mixing time of the giant component of a random graph

2014 ◽  
Vol 45 (3) ◽  
pp. 383-407 ◽  
Author(s):  
Itai Benjamini ◽  
Gady Kozma ◽  
Nicholas Wormald
Keyword(s):  
Random Graph ◽  
Mixing Time ◽  
Giant Component

scholarly journals How to determine if a random graph with a fixed degree sequence has a giant component

2017 ◽  
Vol 170 (1-2) ◽  
pp. 263-310 ◽  
Author(s):  
Felix Joos ◽  
Guillem Perarnau ◽  
Dieter Rautenbach ◽  
Bruce Reed
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Giant Component ◽  
Fixed Degree

scholarly journals Power of $k$ Choices and Rainbow Spanning Trees in Random Graphs

10.37236/4642 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Deepak Bal ◽  
Patrick Bennett ◽  
Alan Frieze ◽  
Paweł Prałat
Keyword(s):  
Stochastic Process ◽  
Random Graph ◽  
Random Graphs ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Special Importance ◽  
Sharp Threshold

We consider the Erdős-Rényi random graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let $\mathcal{G}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\ldots, m$) of $\mathcal{G}(n,m)$ independently chooses precisely $k \in\mathbb{N}$ colours, uniformly at random, from a given set of $n-1$ colours (one may view $e_i$ as a multi-edge). We stop the process prematurely at time $M$ when the following two events hold: $\mathcal{G}(n,M)$ is connected and every colour occurs at least once ($M={n \choose 2}$ if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\mathcal{G}(n,M)$ has a rainbow spanning tree (that is, multicoloured tree on $n$ vertices). Clearly, both properties are necessary for the desired tree to exist.In 1994, Frieze and McKay investigated the case $k=1$ and the answer to this question is "yes" (asymptotically almost surely). However, since the sharp threshold for connectivity is $\frac {n}{2} \log n$ and the sharp threshold for seeing all the colours is $\frac{n}{k} \log n$, the case $k=2$ is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is "yes" also for $k \ge 2$.

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