We consider simple cube-curves in the orthogonal 3D grid. The union of all cells contained in such a curve (also called the tube of this curve) is a polyhedrally bounded set. The curve’s length is defined to be that of the minimum-length polygonal curve (MLP) fully contained and complete in the tube of the curve. So far no provable general algorithm is known for the approximative calculation of...

We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an “edge slide” operation on spanning trees. This gives an answer in the case n = 3 to a question implicitly raised by Stanley. Our argument also gives a bijective proof of the n = 3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.

Let ) (m max G ξ denote the maximum number of edges in a subgraph of graph G induced by m nodes. The n -dimensional augmented cube, denoted as n AQ , a variation of the hypercube, possesses some properties superior to those of the hypercube. We study the cases when G is the augmented cube n AQ . In this paper, we show that i p i r i n AQ i p m max )2 2 1 2 ( = ) ( 0 = − + ∑ ξ , where r p p p > ...

It iswell known that all n×n partial Latin squareswith atmost n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes.We show that if an n×n×n partial Latin cube contains at most n − 1 entries, no two of which occupy the same row, then the partial Latin cube is completable. Also included in this paper is the problem of completing 2×n×n partial Latin...

The Fibonacci cube of dimension n, denoted as Γn, is the subgraph of n-cube Qn induced by vertices with no consecutive 1’s. We study the maximum number of disjoint subgraphs in Γn isomorphic to Qk, and denote this number by qk(n). We prove several recursive results for qk(n), in particular we prove that qk(n) = qk−1(n − 2) + qk(n − 3). We also prove a closed formula in which qk(n) is given in t...

In this paper, we present a strongly fault-tolerant design for the k-ary n-cube multiprocessor and examine its reconfigurability. Our design augments the k-ary n-cube with ðjÞ n spare nodes. Each set of j regular nodes is connected to a spare node and the spare nodes are interconnected as either a ðkjÞ-ary n-cube if j 61⁄4 k2 or a hypercube of dimension n if j 1⁄4 k2 . Our approach utilizes the...

The twisted cube TQn, is derived by changing some connection of hypercube Qn according to specific rules. Recently, many topological properties of this variation cube are studied. In this paper, we consider a faulty twisted n-cube with both edge and/or node faults. Let F be a subset of V(TQn) 5 E(TQn), we prove that TQn −F remains hamiltonian if |F| [ n−2. Moreover, we prove that there exists a...

Percolation with edge-passage probability p and first-passage percolation are studied for the n-cube Bn = {0, 1}n with nearest neighbor edges. For oriented and unoriented percolation, p = e/n and p = 1/n are the respective critical probabilities. For oriented first-passage percolation with i.i.d. edge-passage times having a density of 1 near the origin, the percolation time (time to reach the o...

Let ${\mit\Theta} ^{(n)}$ be a random vector uniformly distributed on the unit sphere $\mathbb S ^{n-1}$ in R^n$. Consider projection of uniform distribution cube $[-1,1]^n$ to line spanned by ^{(n)}$. The project

We study a family of graphs related to the n-cube. The middle cube graph of parameter k is the subgraph of Q2k−1 induced by the set of vertices whose binary representation has either k − 1 or k number of ones. The middle cube graphs can be obtained from the wellknown odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theor...