Related papers: An approximate version of Sidorenko's conjecture

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be…

Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a…

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k+1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k)…

We show that if pn >> log n, the binomial random graph G_{n,p} has an approximate Hamilton decomposition. More precisely, we show that in this range G_{n,p} contains a set of edge-disjoint Hamilton cycles covering almost all of its edges.…

We conjecture that the distribution of the edge-disjoint union of two random regular graphs on the same vertex set is asymptotically equivalent to a random regular graph of the combined degree, provided it grows as the number of vertices…

We study two variations of the Gyarfas--Lehel conjecture on the minimum number of monochromatic components needed to cover an edge-coloured complete bipartite graph. Specifically, we show the following. - For p>> (\log n/n)^{1/2},…

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges…

Let the Andr\'{a}sfai graph $\mathrm{And}_k$ be defined as the graph with vertex set $\{v_0,v_1,...c, v_{3k-2}\}$ and two vertices $v_i$ and $v_j$ being adjacent iff $|i-j| \equiv 1\mod 3$. The graphs $\mathrm{And}_k$ are maximal…

Let $G = (V, E)$ be a graph and $\sigma(G)$ the number of independent (vertex) sets in $G$. Then the Merrifield-Simmons conjecture states that the sign of the term $\sigma(G_{-u}) \cdot \sigma(G_{-v}) - \sigma(G) \cdot \sigma(G_{-u-v})$…

A famous conjecture of Erd\H{o}s and S\'os states that every graph with average degree more than $k - 1$ contains all trees with $k$ edges as subgraphs. We prove that the Erd\H{o}s-S\'os conjecture holds approximately, if the size of the…

The well-known regularity lemma of E. Szemer\'edi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It…

In 2004, Karo\'nski, \L uczak and Thomason proposed $1$-$2$-$3$-conjecture: For every nice graph $G$ there is an edge weighting function $ w:E(G)\rightarrow\{1,2,3\} $ such that the induced vertex coloring is proper. After that, the total…

The Tur\'{a}n number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^{\alpha})$…

A classical theorem of Dirac from 1952 asserts that every graph on $n$ vertices with minimum degree at least $\lceil n/2 \rceil$ is Hamiltonian. In this paper we extend this result to random graphs. Motivated by the study of resilience of…

For a fixed bipartite graph H and given number c, 0<c<1, we determine the threshold T_H(c) which guarantees that any n-vertex graph with at edge density at least T_H(c) contains $(1-o(1))c/v(H) n$ vertex-disjoint copies of H. In the proof…