Related papers: Counting strongly connected $(k_1,k_2)$-directed c…

A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is…

We prove that for $k+1\geq 3$ and $c>(k+1)/2$ w.h.p. the random graph on $n$ vertices, $cn$ edges and minimum degree $k+1$ contains a (near) perfect $k$-matching. As an immediate consequence we get that w.h.p. the $(k+1)$-core of $G_{n,p}$,…

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k,…

A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a…

For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz, Krivelevich and Tyomkyn formulated the following conjecture: When sampling a $k$-vertex subset uniformly at random from a very large graph $G$, then the…

We show that if a graph is k-edge-connected, and we adjoin to it another graph satisfying a "contracted diameter less or equal to 2" condition, with minimal degree greater or equal to k, and some natural hypothesis on the edges connecting…

We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on…

A connected graph $G$ is called strongly Menger (edge) connected if for any two distinct vertices $x,y$ of $G$, there are $\min \{{\rm deg}_G(x), {\rm deg}_G(y)\}$ vertex(edge)-disjoint paths between $x$ and $y$. In this paper, we consider…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of…

The recursive removal of leaves (dead end vertices) and their neighbors from an undirected network results, when this pruning algorithm stops, in a so-called core of the network. This specific subgraph should be distinguished from…

Motivated by the scaling limits of the connected components of the configuration model, we study uniform connected multigraphs with fixed degree sequence $\mathcal{D}$ and with surplus $k$. We call those random graphs…

We study thresholds for the appearance of a 2-core in random hypergraphs that are a mixture of a constant number of random uniform hypergraphs each with a linear number of edges but with different edge sizes. For the case of two overlapping…

Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $\delta\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot…

The minimum semi-degree of a digraph D is the minimum of its minimum outdegree and its minimum indegree. We show that every sufficiently large digraph D with minimum semi-degree at least n/2 +k-1 is k-linked. The bound on the minimum…

For positive integers $n$ and $k$ with $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph with vertex set consisting of all $k$-sets of $\{1,\dots,n\}$, where two $k$-sets are adjacent exactly when they are disjoint. The independent sets…

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1…

A graph $G$ is $m$-joined if there is an edge between every two disjoint $m$-sets of vertices. In this paper, we prove that for any $\varepsilon>0$ and sufficiently large $m, n\in \mathbb{N}$ with $m \le n^{1-\varepsilon}$, every $n$-vertex…

Let $D$ be a digraph. A $k$-container of $D$ between $u$ and $v$, $C(u,v)$, is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u,v)$ of $D$ is a strong (resp. weak) $k^{*}$-container if there is a set of $k$…

The \emph{maximal $k$-edge-connected subgraphs} problem is a classical graph clustering problem studied since the 70's. Surprisingly, no non-trivial technique for this problem in weighted graphs is known: a very straightforward…