Related papers: Barak-Erd\H{o}s graphs and the infinite-bin model

Erd\H{o}s and P\'{o}sa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in…

Start with a graph with a subset of vertices called {\it the border}. A particle released from the origin performs a random walk on the graph until it comes to the immediate neighbourhood of the border, at which point it joins this subset…

We investigate the evidence/flexibility (i.e., "Occam") paradigm and demonstrate the theoretical and empirical consistency of Bayesian evidence for the task of determining an appropriate generative model for network data. This model…

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding…

The Erd\H{o}s-S\'os conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. Despite significant interest, the conjecture remains unsolved. Recently, Caro,…

We improve the best known upper bound on the number of edges in a unit-distance graph on $n$ vertices for each $n\in\{16,\ldots,30\}$. When $n\leq 21$, our bounds match the best known lower bounds, and we fully enumerate the densest…

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarr\`{e}s [Ann. Probab. (2007), to appear], that the attracting edge emerges…

A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture…

While investigating odd-cycle free hypergraphs, Gy\H{o}ri and Lemons introduced a colored version of the classical theorem of Erd\H{o}s and Gallai on $P_k$-free graphs. They proved that any graph $G$ with a proper vertex coloring and no…

We define an inhomogeneous percolation model on "ladder graphs" obtained as direct products of an arbitrary graph $G = (V,E)$ and the set of integers $\mathbb{Z}$ (vertices are thought of as having a "vertical" component indexed by an…

In an undirected graph, the odd cycle packing number is the maximum number of pairwise vertex-disjoint odd cycles. The odd cycle transversal number is the minimum number of vertices that hit every odd cycle. The maximum ratio between…

We prove distance bounds for graphs possessing positive Bakry-\'Emery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits…

For the simple Bratteli diagrams B where there is a single edge connecting any two vertices in consecutive levels, we show that a random order has uncountably many infinite paths if and only if the growth rate of the level-n vertex sets is…

A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times…

For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the…

We use a variant of Bukh's random algebraic method to show that for every natural number $k \geq 2$ there exists a natural number $\ell$ such that, for every $n$, there is a graph with $n$ vertices and $\Omega_k(n^{1 + 1/k})$ edges with at…

Kernel and linear regression have been recently explored in the prediction of graph signals as the output, given arbitrary input signals that are agnostic to the graph. In many real-world problems, the graph expands over time as new nodes…

We study a model of random graphs where each edge is drawn independently (but not necessarily identically distributed) from the others, and then assigned a random weight. When the mean degree of such a graph is low, it is known that the…

The upper tail problem in a sparse Erd\H{o}s-R\'enyi graph asks for the probability that the number of copies of some fixed subgraph exceeds its expected value by a constant factor. We study the analogous problem for oriented subgraphs in…

We consider a growing planar network where a tip grows at constant speed, branches at constant rate and inactivates when it meets a branch already created. We only consider here orthogonal branching occurring always in the same direction.…