Related papers: The diameter of randomly twisted hypercubes

In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there…

A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erd\H{o}s and…

We consider the complete graph $\cK_n$ on $n$ vertices with exponential mean $n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]} C_{ij}/\log{n}$…

In this paper, we derive the exact formula for the probability that three randomly and uniformly selected points from the interior of the unit cube form vertices of an obtuse triangle.

We show that the intrinsic diameter of mean curvature flow in $\mathbb{R}^3$ is uniformly bounded as one approaches the first singular time $T$. This confirms the bounded diameter conjecture of Haslhofer. In addition, we establish several…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence.…

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. $d$-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since…

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the…

We consider a sequence H_N of Hilbert spaces of dimensions d_N tending to infinity. The motivating examples are eigenspaces or quasi-mode spaces of a Laplace or Schrodinger operator. We define a random ONB of H_N by fixing one ONB and…

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…

Structure connectivity and substructure connectivity are innovative indicators for assessing network reliability and fault tolerance. Similarly, fault diameter evaluates fault tolerance and transmission delays in networks. This paper…

The computation of the diameter is one of the most central problems in distributed computation. In the standard CONGEST model, in which two adjacent nodes can exchange $O(\log n)$ bits per round (here $n$ denotes the number of nodes of the…

We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$…

We study Hamiltonicity in random subgraphs of the hypercube $\mathcal{Q}^n$. Our first main theorem is an optimal hitting time result. Consider the random process which includes the edges of $\mathcal{Q}^n$ according to a uniformly chosen…

We consider the standard first passage percolation model in $\ZZ^d$ for $d\geq 2$ and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a…

Consider a $d$-dimensional closed ball $B$ whose center coincides with that of the hypercube $[0,1]^d$. Pick the radius of $B$ in such a way that the vertices of the hypercube are outside of $B$ and the midpoints of its edges in the…

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, that is when p=1/n+ lambda*n^{-4/3}, for some fixed lambda in R. Then, as a metric space with the graph distance rescaled by n^{-1/3}, the sequence of connected…

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional twisted cube, denoted by $TQ_n$, an important variation of the hypercube, possesses some…

We prove a limit theorem for the the maximal interpoint distance (also called the diameter) for a sample of n i.i.d. points in the unit ball of dimension 2 or more. The exact form of the limit distribution and the required normalisation are…