Related papers: Percolation with small clusters on random graphs
Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. Let S be the set of all quadruples (d_0,d_1,d_2,d_3) that are arbitrary close to 3-vertex…
Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…
We prove that for every non-trivial hereditary family of graphs ${\cal P}$ and for every fixed $p \in (0,1)$, the maximum possible number of edges in a subgraph of the random graph $G(n,p)$ which belongs to ${\cal P}$ is, with high…
We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ \Omega\left( \sqrt{\frac{n…
We study the following question raised by Erd\H{o}s and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of…
We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…
We study the problem of efficiently finding large common induced subgraphs of two independent Erd\H{o}s--R\'enyi random graphs $G_1, G_2 \sim \mathbb{G}(n,1/2)$. Recently, Chatterjee and Diaconis showed that the largest common induced…
We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical…
We give nearly optimal bounds on the sample complexity of $(\widetilde{\Omega}(\epsilon),\epsilon)$-tolerant testing the $\rho$-independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random…
An old problem raised independently by Jacobson and Sch\"onheim asks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this…
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…
A well-known theorem of R\"odl says that for every graph $H$, and every $\epsilon>0$, there exists $\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \delta|G|$ such that…
We suggest two related conjectures dealing with the existence of spanning irregular subgraphs of graphs. The first asserts that any $d$-regular graph on $n$ vertices contains a spanning subgraph in which the number of vertices of each…
We define a proportionally dense subgraph (PDS) as an induced subgraph of a graph with the property that each vertex in the PDS is adjacent to proportionally as many vertices in the subgraph as in the graph. We prove that the problem of…
We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution…
The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and…
We bound the second eigenvalue of random $d$-regular graphs, for a wide range of degrees $d$, using a novel approach based on Fourier analysis. Let $G_{n, d}$ be a uniform random $d$-regular graph on $n$ vertices, and let $\lambda (G_{n,…
We prove that the inducibility of $P_4$ in ordered monotone balanced bipartite graphs is $2/e^2$, establishing the smallest known graph with transcendental Tur\'an-type density. Moreover, the limit object is a binary graphon, so it…
We study site and bond percolation on directed simple random graphs with a given degree distribution and derive the expressions for the critical value of the percolation probability above which the giant strongly connected component emerges…
We show that there exist constants $\delta_1,\delta_2>0$ such that if $G$ is an $(n,d,\lambda)$-graph with $\lambda/d\le\delta_1$, then $G$ contains an induced cycle of length at least $\delta_2n/d$. We further demonstrate that, up to a…