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This paper studies the problem of detecting the presence of a small dense community planted in a large Erd\H{o}s-R\'enyi random graph $\mathcal{G}(N,q)$, where the edge probability within the community exceeds $q$ by a constant factor.…
We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection…
In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…
Conditional lower bounds for dynamic graph problems has received a great deal of attention in recent years. While many results are now known for the fully-dynamic case and such bounds often imply worst-case bounds for the partially dynamic…
We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
The study of domination in graphs has led to a variety of domination problems studied in the literature. Most of these follow the following general framework: Given a graph $G$ and an integer $k$, decide if there is a set $S$ of $k$…
The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$…
We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…
Let $N$ be the number of triangles in an Erd\H{o}s-R\'enyi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean…
The \textsc{Bipartite Contraction} problem is to decide, given a graph $G$ and a parameter $k$, whether we can can obtain a bipartite graph from $G$ by at most $k$ edge contractions. The fixed-parameter tractability of the problem was shown…
In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the…
In multi-task learning (MTL) with each task involving graph-dependent data, existing generalization analyses yield a \emph{sub-optimal} risk bound of $O(\frac{1}{\sqrt{n}})$, where $n$ is the number of training samples of each task.…
In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model $\Gnd$ for $d=o(n^{1/5})$ is concentrated in two consecutive values, thus extending a previous result of Achlioptas…
Probabilistic message-passing algorithms are developed for routing transmissions in multi-wavelength optical communication networks, under node and edge-disjoint routing constraints and for various objective functions. Global routing…
In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph $G$, each pair of vertices are joined by an edge with a probability $p$, where $p$ is a…
We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint…
Semi-random processes involve an adaptive decision-maker, whose goal is to achieve some predetermined objective in an online randomized environment. They have algorithmic implications in various areas of computer science, as well as…
Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to…
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs…