Related papers: The Peculiar Phase Structure of Random Graph Bisec…

A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of $G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at most $k-2$…

We show that by restricting the degrees of the vertices of a graph to an arbitrary set \( \Delta \), the threshold point $ \alpha(\Delta) $ of the phase transition for a random graph with $ n $ vertices and $ m = \alpha(\Delta) n $ edges…

In communication field, an important issue is to group users and base stations to as many as possible subnetworks satisfying certain interference constraints. These problems are usually formulated as a graph partition problems which…

In a graph $G=(V,E)$, a bisection $(X,Y)$ is a partition of $V$ into sets $X$ and $Y$ such that $|X|\le |Y|\le |X|+1$. The size of $(X,Y)$ is the number of edges between $X$ and $Y$. In the Max Bisection problem we are given a graph…

We consider the problem of adding a fixed number of new edges to an undirected graph in order to minimize the diameter of the augmented graph, and under the constraint that the number of edges added for each vertex is bounded by an integer.…

We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…

We report on the phase transition of finding a complete subgraph, of specified dimensions, in a bipartite graph. Finding a complete subgraph in a bipartite graph is a problem that has growing attention in several domains, including…

Given an undirected, unweighted graph with $n$ vertices and $m$ edges, the maximum cut problem is to find a partition of the $n$ vertices into disjoint subsets $V_1$ and $V_2$ such that the number of edges between them is as large as…

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question…

We study the problem of sketching an input graph, so that given the sketch, one can estimate the weight of any cut in the graph within factor $1+\epsilon$. We present lower and upper bounds on the size of a randomized sketch, focusing on…

Graphs are a natural representation of data from various contexts, such as social connections, the web, road networks, and many more. In the last decades, many of these networks have become enormous, requiring efficient algorithms to cut…

We present near-optimal algorithms for detecting small vertex cuts in the CONGEST model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete,…

We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…

We study the joint components in a random `double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices, which contains both a red and a blue spanning…

Consider a critical random multigraph $\mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second…

We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at…

In the graph clustering problem with a planted solution, the input is a graph on $n$ vertices partitioned into $k$ clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce…

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…

We consider the problem of constructing a graph of minimum degree $k\ge 1$ in the following controlled random graph process, introduced recently by Frieze, Krivelevich and Michaeli. Suppose the edges of the complete graph on $n$ vertices…

We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For $\Omega \subset D = (0,1)^d$, with $d \geq 2$, we are given $n$ random i.i.d. points on $D$ whose membership in $\Omega$ is known. We…