scholarly journals Maximal Planar Subgraphs of Fixed Girth in Random Graphs

10.37236/7114 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Manuel Fernández ◽  
Nicholas Sieger ◽  
Michael Tait
Keyword(s):  
Random Graphs ◽  
Similar Threshold ◽  
Planar Subgraphs

In 1991, Bollobás and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$. 

scholarly journals Spanning maximal planar subgraphs of random graphs

1991 ◽  
Vol 2 (2) ◽  
pp. 225-231 ◽  
Author(s):  
B. Bollobás ◽  
A. M. Frieze
Keyword(s):  
Random Graphs ◽  
Planar Subgraphs

scholarly journals Balanced Online Ramsey Games in Random Graphs

10.37236/100 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Anupam Prakash ◽  
Reto Spöhel ◽  
Henning Thomas
Keyword(s):  
Large Class ◽  
Random Graphs ◽  
Arbitrary Number ◽  
Vertex Coloring ◽  
Empty Graph ◽  
Threshold Functions ◽  
Current Graph ◽  
Similar Threshold ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph on $n$ vertices, in every step $r$ new edges are drawn uniformly at random and inserted into the current graph. These edges have to be colored immediately with $r$ available colors, subject to the restriction that each color is used for exactly one of these edges. The player's goal is to avoid creating a monochromatic copy of some fixed graph $F$ for as long as possible. We prove explicit threshold functions for the duration of this game for an arbitrary number of colors $r$ and a large class of graphs $F$. This extends earlier work for the case $r=2$ by Marciniszyn, Mitsche, and Stojaković. We also prove a similar threshold result for the vertex-coloring analogue of this game.

Random Graphs

1998 ◽  
Author(s):  
V. F. Kolchin
Keyword(s):  
Random Graphs

Applications of random graphs

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Social Networks ◽  
Random Graphs ◽  
Interaction Network ◽  
Power Grids ◽  
Protein Protein Interaction ◽  
Topological Features ◽  
First Case ◽  
The Stability ◽  
Protein Protein Interaction Network

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.

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.

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