random subgraphs
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Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Antonio Girão ◽  
Daniela Kühn ◽  
Deryk Osthus
Keyword(s):  

2021 ◽  
pp. 221-227
Author(s):  
Nikita Derevyanko ◽  
Mikhail Koshelev ◽  
Andrei Raigorodskii

Author(s):  
Joshua Erde ◽  
Mihyun Kang ◽  
Michael Krivelevich

Abstract Let G be a graph of minimum degree at least k and let G p be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that G p contains when p = (1 + ε)/k with ε > 0. We show that with high probability G p contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.


2019 ◽  
Vol 12 (7) ◽  
pp. 1153-1161
Author(s):  
Arran Hamm ◽  
Kristen Melton

10.37236/6761 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Stefan Ehard ◽  
Felix Joos

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the $G(n,p)$-model. We show that several results concerning the length of the longest path/cycle naturally translate to $G_p$ if $G$ is an arbitrary graph of minimum degree at least $n-1$.For a constant $c>0$ and $p=\frac{c}{n}$, we show that asymptotically almost surely the length of the longest path in $G_p$ is at least $(1-(1+\epsilon(c))ce^{-c})n$ for some function $\epsilon(c)\to 0$ as $c\to \infty$, and the length of the longest cycle is a least $(1-O(c^{- \frac{1}{5}}))n$. The first result is asymptotically best-possible. This extends several known results on the length of the longest path/cycle of a random graph in the $G(n,p)$-model to the random graph model $G_p$ where $G$ is a graph of minimum degree at least $n-1$.


2017 ◽  
Vol 222 (1) ◽  
pp. 317-331 ◽  
Author(s):  
Noga Alon ◽  
Alexey Pokrovskiy ◽  
Benny Sudakov

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