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Let r be an integer, f(n) a function, and H a graph. Introduced by Erd\H{o}s, Hajnal, S\'{o}s, and Szemer\'edi, the r-Ramsey-Tur\'{a}n number of H, RT_r(n, H, f(n)), is defined to be the maximum number of edges in an n-vertex, H-free graph…
Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.…
A subset of the integer planar grid $[N] \times [N]$ is called corner-free if it contains no triple of the form $(x,y), (x+\delta,y), (x,y+\delta)$. It is known that such a set has a vanishingly small density, but how large this density can…
Preferential attachment lies at the heart of many network models aiming to replicate features of real world networks. To simulate the attachment process, conduct statistical tests, or obtain input data for benchmarks, efficient algorithms…
Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$…
Despite advances in generative methods, accurately modeling the distribution of graphs remains a challenging task primarily because of the absence of predefined or inherent unique graph representation. Two main strategies have emerged to…
In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$…
We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le…
We study the problem of detecting local geometry in random graphs. We introduce a model $\mathcal{G}(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph on $\mathbb{S}^{d-1}$, while all…
A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general…
Recent years are characterized by an unprecedented quantity of available network data which are produced at an astonishing rate by an heterogeneous variety of interconnected sensors and devices. This high-throughput generation calls for the…
We obtain several sharp spectral bounds, approximations, and exact values for the isoperimetric number and related edge-expansion parameters of graphs. Our results focus on graph powers and on families of graphs with rich algebraic or…
To each graph on $n$ vertices there is an associated subspace of the $n \times n$ matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally…
Many important theorems in combinatorics, such as Szemer\'edi's theorem on arithmetic progressions and the Erd\H{o}s-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In…
We give a simple and natural (probabilistic) construction of hypergraph regularization. It is done just by taking a constant-bounded number of random vertex samplings only one time (thus, iteration-free). It is independent from the…
Symmetric edge polytopes are lattice polytopes associated with finite simple graphs that are of interest in both theory and applications. We investigate the facet structure of symmetric edge polytopes for various models of random graphs.…
We consider the problem of learning the structure of ferromagnetic Ising models Markov on sparse Erdos-Renyi random graph. We propose simple local algorithms and analyze their performance in the regime of correlation decay. We prove that an…
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in…
Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be a prime power. We construct a bipartite, $q$-regular, edge-transitive graph $C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the the…
Fatgraphs are multigraphs enriched with a cyclic order of the edges incident to a vertex. This paper presents algorithms to: (1) generate the set of all fatgraphs having a given genus and number of boundary cycles; (2) compute automorphisms…