Related papers: The random k-matching-free process

We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular…

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower…

We study the k-core of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree…

In [Amir et al.], the authors consider the generalization $\Gor$ of the Erd\H{o}s-R\'enyi random graph process $G$, where instead of adding new edges uniformly, $\Gor$ gives a weight of size 1 to missing edges between pairs of isolated…

In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\"{o}nig's work in 1916 (here $m=nd$ is the…

For a fixed integer $k\geqslant 2$, let $G\in \mathcal{G}(n,p)$ be a simple connected graph on $n\rightarrow\infty$ vertices with the expected degree $d=np$ satisfying $d\geqslant c$ and $d^{k-1}= o(n)$ for some large enough constant $c$.…

The areas of Ramsey theory and random graphs have been closely linked ever since Erd\H{o}s' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was…

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main…

We consider a natural variant of the Erd\H{o}s-R\'enyi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by…

Each vertex of an arbitrary simple graph on $n$ vertices chooses $k$ random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the…

Let $G$ be a graph on $n$ vertices and let $k$ be a fixed positive integer. We denote by $\mathcal G_{\text{$k$-out}}(G)$ the probability space consisting of subgraphs of $G$ where each vertex $v\in V(G)$ randomly picks $k$ neighbors from…

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then…

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…

A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have…

We investigate random processes for generating task-dependency graphs of order $n$ with $m$ edges and a specified number of initial vertices and terminal vertices. In order to do so, we consider two random processes for generating…

The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each…

We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labeled $k$-vertex graphs into itself ($k$…

We describe a new family of $k$-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree $2$, even when the edge density (number of edges…

Differential Privacy is the gold standard in privacy-preserving data analysis. This paper addresses the challenge of producing a differentially edge-private vertex coloring. In this paper, we present two novel algorithms to approach this…

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every…