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We present a method to project a hypercube of arbitrary dimension on the plane, in such a way as to preserve, as well as possible, the distribution of distances between vertices. The method relies on a Montecarlo optimization procedure that…
We study the polyhedral properties of three problems of constructing an optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the first problem we consider a balanced biclique with the same number of vertices in both…
The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be…
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,…
We study the two-dimensional hierarchical rectangle packing problem, motivated by applications in analog integrated circuit layout, facility layout, and logistics. Unlike classical strip or bin packing, the dimensions of the container are…
In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an $n \times…
A hedge graph is a graph whose edge set has been partitioned into groups called hedges. Here we consider a generalization of the well-known \textsc{Cluster Deletion} problem, named \textsc{Hedge Cluster Deletion}. The task is to compute the…
One unsolved mathematical problem remains the perfect cuboid problem. A perfect cuboid is a rectangular parallelepiped whose edges, face diagonals and space diagonal are all expressed as integers. No such cuboid has yet been discovered and…
As a variant of the well-known hypercube, the balanced hypercube $BH_n$ was proposed as a novel interconnection network topology for parallel computing. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2\}$ and $T=\{t_1,t_2\}$…
We consider a swarm of $n$ robots in \mathbb{R}^d. The robots are oblivious, disoriented (no common coordinate system/compass), and have limited visibility (observe other robots up to a constant distance). The basic formation task gathering…
Creating materials with structure that is independently controllable at a range of scales requires breaking naturally occurring hierarchies. Breaking these hierarchies can be achieved via the decoupling of building block attributes from…
Aims: Try to prove the $n$-dimensional balanced hypercube $BH_n$ is $(2n-2)$-fault-tolerant-prescribed hamiltonian laceability. Methods: Prove it by induction on $n$. It is known that the assertation holds for $n\in\{1,2\}$. Assume it holds…
Sequential Latin hypercube designs have recently received great attention for computer experiments. Much of the work has been restricted to invariant spaces. The related systematic construction methods are inflexible while algorithmic…
We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the…
The general d-dimensional twisted group lattice is solved. The irreducible representations of the corresponding group are constructed by an explicit procedure. It is proven that they are complete. All matrix representation solutions to the…
The separation dimension of a hypergraph $G$ is the smallest natural number $d$ for which there is an embedding of $G$ into $\mathbb{R}^d$, such that any pair of disjoint edges is separated by some hyperplane normal to one of the axes. The…
Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny \neq 0$, $d\geq…
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
The Ordered Covering Problem (OCP) arises in the context of the Discretizable Molecular Distance Geometry Problem (DMDGP), where the ordering of pruning edges significantly impacts the performance of the SBBU algorithm for protein structure…
We introduce in this paper an exact solvable BCS-Hubbard model in arbitrary dimensions. The model describes a p-wave BCS superconductor with equal spin pairing moving on a bipartite (cubic, square etc.) lattice with on site Hubbard…