We investigate the function dA(n), which gives the size of a least size generating set for A, in the case where A has a cube term. We show that if A has a k-cube term and A is finitely generated, then dA(n) ∈ O(log(n)) if A is perfect and dA(n) ∈ O(n) if A is imperfect. When A is finite, then one may replace “Big Oh” with “Big Theta” in these estimates.

We study random subgraphs of the n-cube {0, 1}n, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected cluster size of a fixed vertex attains the value λ2n/3, where λ is a small positive constant. In two previous papers, we developed a general theory of random subgraphs of high-dimensional finite connected transitive graphs. In this pap...

The Fibonacci cube Γn is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λn is obtained from Γn by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γn and Λn is ∑k i=0 ( n−2i k−i )( i+1 n−k−i+1 ) and ∑k i=0 [ 2 ( i 2i+k−n )( n−2i−1 k−i ) + ( i−1 2i+k−n )( n−2i k−i )] , respectively. ...

We examine the issue of running algorithms on a hypercube which has both node and edge faults, and we assume a worst case distribution of the faults. We prove that for any constant c, an n-dimensional hypercube (n-cube) with n c faulty components contains a fault-free subgraph that can implement a large class of hypercube algorithms with only a constant factor slowdown. In addition, our approac...

A range-max query finds the maximum value over all selected cells of an on-line analytical processing (OLAP) data cube where the selection is specified by ranges of contiguous values for each dimension. One of the approaches to process such queries is to precompute a prefix cube (PC), which is a cube of the same dimensionality and size as the original data cube, but with some pre-computed resul...

The dual-cube is a newly proposed topology for interconnection networks, which uses low dimensional hypercubes as building blocks. The primary advantages of the dual-cube over the hypercube are that, with the same node degree n, the dual-cube contains 2 1 times more nodes than the hypercube and, with the same amount of nodes, the dual-cube has approximately 50% less links than the hypercube. Th...

In this paper a parallel algorithm for solving systems of linear equation on the kary n-cube is presented and evaluated for the first time. The proposed algorithm is of O(N/k) computation complexity and uses O(Nn) communication time to factorize a matrix of order N on the k-ary n-cube. This is better than the best known results for the hypercube, O(N log k), and the mesh, O(N √ kn), each with a...

Generalizing a result of Miyakawa, Nozaki, Pogosyan and Rosenberg, we prove that there is a one-to-one correspondence between the set of intersecting antichains in a subset of the lower half of the kvalued n-cube and the set of intersecting antichains in the k-valued (n− 1)-cube.

An n-bit gray code can be represented as a hamilton cycle in the n-cube together with a direction. Two n-bit gray codes are equivalent if there is an automorphism of the n-cube that maps the edge set of one onto the other. In this paper we detail a method for counting the number of equivalence classes of 5-bit gray codes, and classify the equivalence classes according to the distribution of the...

Consider a hypergraph $H_n^d$ where the vertices are points of $d$-dimensional combinatorial cube $n^d$ and edges all sets $n$ such that they in one line. We study structure group automorphisms $H_n^d$, i.e., permutations preserving edges. In this paper we provide complete characterization. Moreover, consider Colored Cube Isomorphism problem deciding whether for two colorings there exists an au...