Related papers: Improved Upper Bounds for Slicing the Hypercube

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower…

For $q,n,d \in \mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \leq 65$, $A_4(11,8)\leq 60$ and…

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…

Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ contains at least $m$ points from $P$.…

Simplex slicing (Webb, 1996) is a sharp upper bound on the volume of central hyperplane sections of the regular simplex. We extend this to sharp bounds in the probabilistic framework of negative moments, and beyond, of centred log-concave…

The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional…

We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an $\widetilde{\Omega}(n)$ lower bound for…

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…

In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is…

We show that every triangulation (maximal planar graph) on $n\ge 6$ vertices can be flipped into a Hamiltonian triangulation using a sequence of less than $n/2$ combinatorial edge flips. The previously best upper bound uses $4$-connectivity…

We investigate a very recent concept for visualizing various aspects of a graph in the plane using a collection of drawings introduced by Hlin\v{e}n\'y and Masa\v{r}\'ik [GD 2023]. Formally, given a graph $G$, we aim to find an uncrossed…

Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize…

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the orignial polytope are hereditary to its…

We describe an algorithm for solving an important geometric problem arising in computer-aided manufacturing. When cutting away a region from a solid piece of material -- such as steel, wood, ceramics, or plastic -- using a rough tool in a…

We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges,…

Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show…

Given an integer $1\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\leq i \leq j$, have color $1$ and all other edges have…

As the scale of data centers continues to grow, there is an increasing demand for interconnection networks to resist malicious attacks. Hence, it is necessary to evaluate the reliability of networks under various fault patterns. The family…

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an…

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…