This paper considers the following game on a hypercube, first suggested by Lagarias and Saks. Suppose 2 pebbles are distributed onto vertices of an n-cube (with 2 vertices). A pebbling step is to remove two pebbles from some vertex and then place one pebble at an adjacent vertex. The question of interest is to determine if it is possible to get one pebble to a specified vertex by repeatedly usi...

In this paper a method of embedding a linear graph in n-cube is discussed. Necessary and sufficient conditions on node configurations for the embeddability are obtained by analyzing the structure of n-cube in code space with respect to the distance 2c by means of Reed-Muller code. The fundamental idea in this study is to modify a given graph iteratively in such a manner that the connecting prop...

We present a real-time fault-tolerant design for the k-ary n-cube multiprocessor and examine its reconfigurability. The k-ary n-cube is augmented by spare nodes at stages one and two. We consider two modes of operations, one under heavy computation or hard deadline and the other under light computation or soft deadline. We assume that faulty nodes cannot compute, but retain their ability to com...

A collection H of integers is called an affine d-cube if there exist d+ 1 positive integers x0, x1, . . . , xd so that H = { x0 + ∑ i∈I xi : I ⊆ {1, 2, . . . , d} } . We address both density and Ramsey-type questions for affine d-cubes. Regarding density results, upper bounds are found for the size of the largest subset of {1, 2, . . . , n} not containing an affine d-cube. In 1892 Hilbert publi...