Consider lhe rr-cube [0. l]" in R'. This has l" r'ertices and volume I. Pick .\' = .\'(r) r'erlices indcpendentll' a1 random. form their convex hull. and let I'n be its expected volume. Hor"'large should ,\'(rt) be to pick up significant volume? Ler r = 2l"Q x I.213. and let r > 0. We have rc-cenlll' shown that. ds n-t-.1;-0 if .\'(r)< (,(-r)'.and I,-I it.\'(n)>\r+r)'. We discuss this and relat...

Let an edge cut partition the vertex set of the n-cube into k subsets A1, ..., Ak with ||Ai| − |Aj || ≤ 1. We consider the problem to determine minimal size of such a cut and present its asymptotic as n, k → ∞ and also as n → ∞ and k is a constant of the form k = 2a ± 2b with a ≥ b ≥ 0.

This paper introduces a complete description of constructing a Hamiltonian cycle in the Extended OTIS-n-Cube topology. The recently proposed network has many good topological features such as regular degree, semantic structure, low diameter, and ability to embed graphs and cycles. Constructing a Hamiltonian cycle is an important feature for any topology due to the importance of broadcast messag...

Alon, N., Transmitting in the n-dimensional cube, Discrete Applied Mathematics 37/38 (1992) 9-1 I. Motivated by a certain communication problem we show that for any integer n and for any sequence (a,,...,ak) of k = [n/21 binary vectors of length n, there is a binary vector z of length n whose Hamming distance from a, is strictly bigger than k-i for all 1 5 i 5 k. The n-dimensional cube is the g...

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored ncube is said to be balanced if its center of mass coincides with its geometric center. Let Bn,2k be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read, and Robinson conjectured that for n ≥ 1, the se...

This paper confirms a conjecture of Amdeberhan and Moll that the power of 2 dividing the number of plane partitions in an n-cube is greater than the power of 2 dividing the number of totally symmetric plane partitions in the same cube when n is even, and less when n is odd.

A sequence of binary words of length n is called a cube dominating path, if the Hamming distance between two consecutive words is always one, and every binary word of length n is within Hamming distance one from at least one of these words. If also the first and last words are Hamming distance one apart, the sequence is called a cube dominating cycle. Bounds on the cardinality of such sequences...

The hypercube of dimension n is the graph whose vertices are $$2^n$$ binary words length n, and there an edge between two them if they have Hamming distance 1. We consider edit based on swaps mismatches, to which we refer as tilde-distance, define tilde-hypercube with edges linking at tilde-distance Then, introduce study some isometric subgraphs obtained by using special called tilde-isometric ...

The crossed cube is one of the most notable variations of hypercube, but some properties of the former are superior to those of the latter. For example, the diameter of the crossed cube is almost the half of that of the hypercube. In this paper, we focus on the problem embedding a Hamiltonian cycle through an arbitrary given edge in the crossed cube. We give necessary and sufficient condition f...

The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks, including the ring, the torus and the hypercube, that are attractive in both theoretical interests and practical systems can be regarded as the subclasses of the k-ary n-cubes. In this paper, we investigate the fault tolerant cycle e...