The ndimensional hypercube, denoted qn, is the graph on 2n vertices labeled by lengthn. It is a closed, compact, convex figure whose 1 skeleton consists of groups of opposite parallel line segments aligned in each of the spaces dimensions, perpendicular to each other and of the same length. The total number of edges in an ndimensional hypercube is n2n. The problem of efficient computation of maximum matchings in the n dimensional cube, which is applied in coding theory, is solved. Given a graph g, an rcoloring of g is a function c.
The ndimensional hypercube or ncube is a highly concurrent multiprocessor. To construct the edges, we label the vertices 0 to 2n 1 in binary. A ramseytype result for the hypercube stanford cs theory. The n cube q n is the graph whose vertex set is the set of all n dimensional boolean vectors, two vertices being joined if and only if they differ in exactly one coordinate. Lecture 1 department of theoretical computer science and. This method is based on the explicit construction theorem 1 of the maps of the vertex set that induce largest matchings in any bipartite subgraph of the n dimensional cube for any n. If u v e i, the edge uv2eq n is said to have direction i. For this reason, euler is generally hailed as the inventor of graph theory. Definition 1 the ndimensional hypercube qn also known as an ncube, a boolean cube, a discrete cube is an undirected graph with v qn zn. In graph theory, the hypercube graph qn is the graph formed from the vertices and edges of an ndimensional hypercube. In geometry, a hypercube is an ndimensional analogue of a square n 2 and a cube n 3. On cubical graphs it is frequently of interest to represent a.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Let qn denote the graph of the ndimensional hypercube whose vertex set is 0,1 n and two vertices are adjacent if they differ in exactly one coordinate. The simplex is a convex the simplex is a convex hull of n 1 points in a general position e. Hypercube architectures an excerpt from history 198387 cosmic. An extension is the three dimensional kary hamming graph, which for k 2 is the cube graph. N, the complete bipartite graph k m, n is a bipartite graph where v 1 m, v 2 n, and e v 1,v 2v 1. In graph theory, the hypercube graph q n is the graph formed from the vertices and edges of an ndimensional hypercube. Graphs of this sort occur in the theory of parallel processing in computers. The distinguishing number of the augmented cube and hypercube. Every connected graph with at least two vertices has an edge.
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