On random k-out sub-graphs of large graphs

We consider random sub-graphs of a fixed graph $G=(V,E)$ with large minimum degree. We fix a positive integer $k$ and let $G_k$ be the random sub-graph where each $v\in V$ independently chooses $k$ random neighbors, making $kn$ edges in all. When the minimum degree $\delta(G)\geq (\frac12+\epsilon)n,\,n=|V|$ then $G_k$ is $k$-connected w.h.p. for $k=O(1)$; Hamiltonian for $k$ sufficiently large. When $\delta(G) \geq m$, then $G_k$ has a cycle of length $(1-\epsilon)m$ for $k\geq k_\epsilon$. By w.h.p. we mean that the probability of non-occurrence can be bounded by a function $\phi(n)$ (or $\phi(m)$) where $\lim_{n\to\infty}\phi(n)=0$

Random subgraphs make identification affordable

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the identifying code number (the size of a smallest identifying code), which indeed is not even a monotone parameter with respect to graph inclusion. We show that every graph $G$ with $n$ vertices, maximum degree $\Delta=\omega(1)$ and minimum degree $\delta\geq c\log{\Delta}$, for some constant $c>0$, contains a large spanning subgraph which admits an identifying code with size $O\left(\frac{n\log{\Delta}}{\delta}\right)$. In particular, if $\delta=\Theta(n)$, then $G$ has a dense spanning subgraph with identifying code $O\left(\log n\right)$, namely, of asymptotically optimal size. The subgraph we build is created using a probabilistic approach, and we use an interplay of various random methods to analyze it. Moreover we show that the result is essentially best possible, both in terms of the number of deleted edges and the size of the identifying code

The probability of planarity of a random graph near the critical point

Erdős and Rényi conjectured in 1960 that the limiting probability $p$ that a random graph with $n$ vertices and $M=n/2$ edges is planar exists. It has been shown that indeed p exists and is a constant strictly between 0 and 1. In this paper we answer completely this long standing question by finding an exact expression for this probability, whose approximate value turns out to be $p ≈0.99780$. More generally, we compute the probability of planarity at the critical window of width $n^{2/3}$ around the critical point $M=n/2$. We extend these results to some classes of graphs closed under taking minors. As an example, we show that the probability of being series-parallel converges to 0.98003. Our proofs rely on exploiting the structure of random graphs in the critical window, obtained previously by Janson, Łuczak and Wierman, by means of generating functions and analytic methods. This is a striking example of how analytic combinatorics can be applied to classical problems on random graphs

On the Non-Planarity of a Random Subgraph

<p>Let <em>G</em> be a finite graph with minimum degree <em>r</em>. Form a random subgraph <em>G_p </em>of <em>G</em> by taking each edge of <em>G</em> into <em>G_p</em>independently and with probability <em>p</em>. We prove that for any constant ε > 0, if , then <em>G_p </em>is non-planar with probability approaching 1 as <em>r</em> grows. This generalizes classical results on planarity of binomial random graphs.</p