Related papers: Percolation with small clusters on random graphs

Let $\mathcal{C}$ and $\mathcal{D}$ be hereditary graph classes. Consider the following problem: given a graph $G\in\mathcal{D}$, find a largest, in terms of the number of vertices, induced subgraph of $G$ that belongs to $\mathcal{C}$. We…

We prove that for all $0\leq t\leq k$ and $d\geq 2k$, every graph $G$ with treewidth at most $k$ has a `large' induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$…

For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was…

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. We show for $d \geq 2$ that if $\lambda$ is…

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within…

Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterise the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this…

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the…

An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $\alpha$.…

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a…

We show that every graph has an induced pseudoforest of at least $n-m/4.5$ vertices, an induced partial 2-tree of at least $n-m/5$ vertices, and an induced planar subgraph of at least $n-m/5.2174$ vertices. These results are constructive,…

We initiate the systematic study of the following Tur\'an-type question. Suppose $\Gamma$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$…

Given a large graph $H$, does the binomial random graph $G(n,p)$ contain a copy of $H$ as an induced subgraph with high probability? This classical question has been studied extensively for various graphs $H$, going back to the study of the…

We prove that the `Upper Matching Conjecture' of Friedland, Krop, and Markstr\"om and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every…

In $r$-neighbor bootstrap percolation on the vertex set of a graph $G$, a set $A$ of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least $r$ previously infected neighbors. When the…

In this paper, we investigate the induced subgraphs of percolated random geometric graphs, and get some asymptotic results for the expected number of the subgraphs. Moreover, we get the Poisson approximation for the counting by Stein's…

We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erd\"os-Renyi graphs of constant average degree. We prove that the…

We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More…

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations…

Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which…

Bootstrap percolation in (random) graphs is a contagion dynamics among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$…