scholarly journals On the Probability that a Random Subgraph Contains a Circuit

2016 ◽  
Vol 85 (3) ◽  
pp. 644-650
Author(s):  
Peter Nelson
Keyword(s):  
Random Subgraph

Some remarks about extreme degrees in a random graph

1986 ◽  
Vol 100 (1) ◽  
pp. 167-174 ◽  
Author(s):  
Zbigniew Palka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Complete Graph ◽  
Probability Distributions ◽  
Supplementary Information ◽  
Random Subgraph

Let Kn, p be a random subgraph of a complete graph Kn obtained by removing edges, each with the same probability q = 1 – p, independently of all other edges (i.e. each edge remains in Kn, p with probability p). Very detailed results devoted to probability distributions of the number of vertices of a given degree, as well as of the extreme degrees of Kn, p, have already been obtained by many authors (see e.g. [l]–[5], [7]–[9]). A similar subject for other models of random graphs has been investigated in [10]–[13], The aim of this note is to give some supplementary information about the distribution of the ith smallest (i ≥ 1 is fixed) and the ith largest degree in a sparse random graph Kn, p, i.e. when p = p(n) = o(1).

scholarly journals The Spectral Gap of a Random Subgraph of a Graph

2007 ◽  
Vol 4 (2-3) ◽  
pp. 225-244 ◽  
Author(s):  
Fan Chung ◽  
Paul Horn
Keyword(s):  
Spectral Gap ◽  
Random Subgraph

The diameter of a random subgraph of the hypercube

2012 ◽  
Vol 41 (2) ◽  
pp. 282-291
Author(s):  
Tomáš Kulich
Keyword(s):  
Random Subgraph

A Sharp Threshold for Network Reliability

2002 ◽  
Vol 11 (5) ◽  
pp. 465-474 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV ◽  
VAN H. VU
Keyword(s):  
Classical Result ◽  
Network Reliability ◽  
Average Degree ◽  
Edge Connectivity ◽  
Sharp Threshold ◽  
Random Subgraph ◽  
Connectivity Threshold

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) [Gt ] d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

scholarly journals Ramsey Properties of Random Subgraphs of Pseudo-Random Graphs

10.37236/2468 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Jia Shen
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Threshold Function ◽  
Vertex Coloring ◽  
Similar Argument ◽  
Sharp Threshold ◽  
Absolute Value ◽  
Random Subgraph ◽  
Vertex Disjoint ◽  
Random Subgraphs

Let $G=(V,E)$ be a $d$-regular graph of order $n$. Let $G_p$ be the random subgraph of $G$ for which each edge is selected from $E(G)$ independently at random with probability $p$. For a fixed graph $H$, define $m(H):=$max$\{e(H')/(v(H')-1):H' \subseteq H\}$. We prove that $n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex coloring, respectively, provided the eigenvalue $\lambda$ of $G$ that is second largest in absolute value is significantly smaller than $d$.As a consequence, it is also shown that $\displaystyle n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to contain a family of vertex disjoint copies of $H$ (an $H$ packing) that covers $(1-o(1))n$ vertices of $G$. Using a similar argument, the sharp threshold function for $G_p$ to contain $H$ as a subgraph is obtained as well.

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

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