Related papers: Long paths and cycles in random subgraphs of graph…

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

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…

We discuss the length of the longest cycle in a sparse random graph $G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function $f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$…

Let $G$ be any 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 $p$. Recently, Krivelevich, Lee and Sudakov showed that if $pk\to\infty$ then with…

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k +…

Two sharp lower bounds for the length of a longest cycle $C$ of a graph $G$ are presented in terms of the lengths of a longest path and a longest cycle of $G-C$, denoted by $\overline{p}$ and $\overline{c}$, respectively, combined with…

Given a graph $G$ and probability $p$, we form the random subgraph $G_p$ by retaining each edge of $G$ independently with probability $p$. Given $d\in\mathbb{N}$ and constants $0<c<1, \varepsilon>0$, we show that if every subset $S\subseteq…

For a graph $G$, $n$ denotes the order of $G$, $p$ the order of a longest path in $G$ and $c$ the order of a longest cycle. We show that if $G$ is a 2-connected graph such that $d(x)+d(y)+d(z)\ge p+2$ for all triples $x,y,z$ of independent…

Long paths and cycles in sparse random graphs and digraphs were studied intensively in the 1980's. It was finally shown by Frieze in 1986 that the random graph $\cG(n,p)$ with $p=c/n$ has a cycle on at all but at most $(1+\epsilon)ce^{-c}n$…

A graph $G$ on $n$ vertices is \textit{pancyclic} if it contains cycles of length $t$ for all $3 \leq t \leq n$. In this paper we prove that for any fixed $\epsilon>0$, the random graph $G(n,p)$ with $p(n)\gg n^{-1/2}$ asymptotically almost…

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph $G\sim {\mathcal G}(n,p)$ in order to find a subgraph which…

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…

We study the model $G_\alpha\cup G(n,p)$ of randomly perturbed dense graphs, where $G_\alpha$ is any $n$-vertex graph with minimum degree at least $\alpha n$ and $G(n,p)$ is the binomial random graph. We introduce a general approach for…

We study the set ${\cal L}(G)$ of lengths of all cycles that appear in a random $d$-regular $G$ on $n$ vertices for a fixed $d\geq 3$, as well as in Erd\H{o}s--R\'enyi random graphs on $n$ vertices with a fixed average degree $c>1$.…

We study the Tur\'an number of long cycles in random graphs and in pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the…

We consider random temporal graphs, a version of the classical Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a distinct random time stamp, and connectivity is constrained to sequences of edges with increasing time…

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the…

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\epsilon)/n)$ that holds w.h.p. for all $\epsilon=\epsilon(n)$ such that $\epsilon^3n\to \infty$. In the case $\epsilon\leq \epsilon_0$ for…

Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine…

Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length $k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under the assumption…