Related papers: Finding long cycles in a percolated expander graph…

Let $G$ be a vertex-transitive graph of superlinear polynomial growth. Given $r>0$, let $G_r$ be the graph on the same vertex set as $G$, with two vertices joined by an edge if and only if they are at graph distance at most $r$ apart in…

Let $G\sim G(n,p)$ be a (hidden) Erd\H{o}s-R\'enyi random graph with $p=(1+ \varepsilon)/n$ for some fixed constant $ \varepsilon >0$. Ferber, Krivelevich, Sudakov, and Vieira showed that to reveal a path of length…

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…

We say that a graph $G=(V,E)$ on $n$ vertices is a $\beta$-expander for some constant $\beta>0$ if every $U\subseteq V$ of cardinality $|U|\leq \frac{n}{2}$ satisfies $|N_G(U)|\geq \beta|U|$ where $N_G(U)$ denotes the neighborhood of $U$.…

We present exact solutions for the size of the giant connected component (GCC) of graphs composed of higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. We use our theoretical result to find the…

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…

In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…

We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are…

Resolving a conjecture of Bollob\'{a}s and Erd\H{o}s, Gy\'{a}rf\'{a}s proved that every graph $G$ of chromatic number $k+1\geq 3$ contains cycles of $\lfloor\frac{k}{2}\rfloor$ distinct odd lengths. We strengthen this prominent result by…

We consider bootstrap percolation on the binomial random graph $G(n,p)$ with infection threshold $r\in \mathbb{N}$, an infection process which starts from a set of initially infected vertices and in each step every vertex with at least $r$…

In this paper, we study the acquaintance time $\AC(G)$ defined for a connected graph $G$. We focus on $\G(n,r,p)$, a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from…

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 investigate the following vertex percolation process. Starting with a random regular graph of constant degree, delete each vertex independently with probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We show…

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards…

In r-neighbour bootstrap percolation on the vertex set of a graph G, vertices are initially infected independently with some probability p. At each time step, the infected set expands by infecting all uninfected vertices that have at least…

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation…

Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1…

Understanding how the cycles of a graph or digraph behave in general has always been an important point of graph theory. In this paper, we study the question of finding a set of $k$ vertex-disjoint cycles (resp. directed cycles) of distinct…

Let $k \geq 3$ be an integer, $h_{k}(G)$ be the number of vertices of degree at least $2k$ in a graph $G$, and $\ell_{k}(G)$ be the number of vertices of degree at most $2k-2$ in $G$. Dirac and Erd\H{o}s proved in 1963 that if $h_{k}(G) -…