Related papers: Three-Neighbour Bootstrap Percolation in Thin Thre…

We obtain a new lower bound of 0.06576 for the 1-entanglement critical probability (in dimension 3), and prove that the critical point for the existence of a sphere surrounding the origin and intersecting only closed bonds in $\mathbb{Z}^d$…

Percolation in an information-theoretically secure graph is considered where both the legitimate and the eavesdropper nodes are distributed as Poisson point processes. For both the path-loss and the path-loss plus fading model, upper and…

In the random $r$-neighbour bootstrap percolation process on a graph $G$, a set of initially infected vertices is chosen at random by retaining each vertex of $G$ independently with probability $p\in (0,1)$, and "healthy" vertices get…

Distributions of pebbles to the vertices of a graph are said to be solvable when a pebble may be moved to any specified vertex using a sequence of admissible pebbling rules. The optimal pebbling number is the least number of pebbles needed…

Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible…

Consider a uniform expanders family G_n with a uniform bound on the degrees. It is shown that for any p and c>0, a random subgraph of G_n obtained by retaining each edge, randomly and independently, with probability p, will have at most one…

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained,…

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities $p$, $r$, and $t$…

The degree diameter problem asks for the maximum possible number of vertices in a graph of maximum degree $\Delta$ and diameter $D$. In this paper, we focus on planar graphs of diameter $3$. Fellows, Hell and Seyffarth (1995) proved that…

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of two pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of…

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an…

Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled $+$…

We investigate the onset of the discontinuous percolation transition in small-world hyperbolic networks by studying the systems-size scaling of the typical largest cluster approaching the transition, $p\nearrow p_{c}$. To this end, we…

We consider problems that can be formulated as a task of finding an optimal triangulation of a graph w.r.t. some notion of optimality. We present algorithms parameterized by the size of a minimum edge clique cover ($cc$) to such problems.…

Let $[a,b]$ denote the integers between $a$ and $b$ inclusive and, for a finite subset $X \subseteq \mathbb{Z}$, let the diameter of $X$ be equal to $\max(X)-\min(X)$. We write $X<_p\,Y$ provided $\max(X)<\min(Y)$. For a positive integer…

Let $n_g(k)$ denote the smallest order of a $k$-chromatic graph of girth at least $g$. We consider the problem of determining $n_g(k)$ for small values of $k$ and $g$. After giving an overview of what is known about $n_g(k)$, we provide…

Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^d$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,\dots,…

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning…

We give a complete description of the set of triples (a,b,c) of real numbers with the following property. There exists a constant K such that a n_3 + b n_2 + c n_1 - K is a lower bound for the matching number of every connected subcubic…