Related papers: The Z-cubes: a hypercube variant with small diamet…
We study a family of graphs related to the $n$-cube. The middle cube graph of parameter $k$ is the subgraph of $Q_{2k-1}$ induced by the set of vertices whose binary representation has either $k-1$ or $k$ number of ones. The middle cube…
Shape metrics for objects in high dimensions remain sparse. Those that do exist, such as hyper-volume, remain limited to objects that are better understood such as Platonic solids and $n$-Cubes. Further, understanding objects of ill-defined…
Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi\'{c}'s and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the…
The diameter of the Cayley graph of the Rubik's Cube group is the fewest number of turns needed to solve the Cube from the hardest initial configuration. For the 2$\times$2$\times$2 Cube, the diameter is 11 in the half-turn metric, 14 in…
Let $k\geq2$. Then the $k$-th order Fibonacci cube $\Gamma^{(k)}_{n}$ is the subgraph of the hypercube $Q_{n}$ induced by vertices without $k$ consecutive $1$s. The case $k=2$ corresponds to the classic Fibonacci cube $\Gamma_{n}$. There…
The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n…
We prove that at least $\Omega(n^{0.51})$ hyperplanes are needed to slice all edges of the $n$-dimensional hypercube. We provide a couple of applications: lower bounds on the computational complexity of parity, and a lower bound on the…
The Hales numbered $n$-dimensional hypercube and the corresponding adjacency matrix exhibit interesting recursive structures in $n$. These structures lead to a very simple proof of the well-known bandwidth formula for hypercube, whose proof…
Cubic planar $n$-vertex graphs with faces of length at most $6$, e.g., fullerene graphs, have diameter in $\Omega(\sqrt{n})$. It has been suspected, that a similar result can be shown for cubic planar graphs with faces of bounded length.…
A frequency $n$-cube $F^n(4;2,2)$ is an $n$-dimensional $4$-by-...-by-$4$ array filled by $0$s and $1$s such that each line contains exactly two $1$s. We classify the frequency $4$-cubes $F^4(4;2,2)$, find a testing set of size $25$ for…
Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and…
In 1979, Shearer and Kleitman conjectured that there exist $\lfloor n/2 \rfloor+1$ orthogonal chain decompositions of the hypercube $Q_n$, and constructed two orthogonal chain decompositions. In this paper, we make the first non-trivial…
In the year 1990, B\'ela Bollob\'as, Imre Leader and Andrew Radcliffe considered the following combinatorial problem: given three parameters k, n and q, find a set of k vertices in the binary n-cube which contains a maximal number of…
We determine the maximal hyperplane sections of the regular $n$-simplex, if the distance of the hyperplane to the centroid is fairly large, i.e. larger than the distance of the centroid to the midpoint of edges. Similar results for the…
A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions…
We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.
The structure of maximal faces of the cone of completely positive matrices is still not well understood in higher dimensions, mainly due to the lack of a general characterization of extreme exposed rays of the copositive cone beyond small…
A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a…
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…
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…