Say we wanted to get an array of edges (1-faces) in a cube (3-dimensional hypercube). That may have been a confusing way to word it, so here are a few examples: Of Integer Sequences." Referenced on Wolfram|Alpha Hypercube Graph Cite this as:įrom MathWorld-A Wolfram Web Resource.I'm attempting to design an algorithm that, given n, m, and vertices (where n = the dimension of a hypercube, m = the dimension of the faces we're trying to generate, and vertices is an ordered list of vertices in an n-dimensional hypercube), returns an array of arrays of vertices representing m-faces in an n-dimensional hypercube. Skiena,ĭiscrete Mathematics: Combinatorics and Graph Theory with Mathematica. "On the Size of Optimal Binary Codes of Length 9 and Covering "Decomposition of Complete Graphs Into Isomorphic Cubes." J. "A Survey of the Theory of Hypercube Graphs."Ĭomput. "On the Unit DistanceĮmbeddability of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Gardner,ĭoughnuts and Other Mathematical Entertainments. With Known or Bounded Crossing Numbers.". Cambridge, England: Cambridge University Press, p. 161,ġ993. Table of Binary/ternary Mixed Covering Codes." J. Of the NATO Advanced Research Workshop on Cycles and Rays: Basic Structures in FiniteĪnd Infinite Graphs held in Montreal, Quebec, May 3-9, 1987 (Ed. "Three Hamilton Decomposition Problems." University of Western Australia. Įggleton and Guy (1970) claimed to have discovered an upper bound for the graph (2017) showed that domination and totalĭomination numbers of the hypercube graph are related by. 2017)Īnd as of April 2018, values are known only up to (Östergård and Blass 2001, Bertolo et al.Ģ004). One, and repeating until the -hypercube graph has been constructed.ĭetermining the domination number is intrinsically difficult (Azarija et al. Unit translation vectors have been used, so there must be a direction not used before),Ĭonnecting the vertices in the translate with the corresponding vertices in the original One unit in a direction not chosen in any of the steps before (only finitely many
Of the square graph, translating the embedding by This canīe established by induction for the -hypercube graph by starting with the unit-distance embedding (Gerbracht 2008), as illustrated above for the first few hypercube graphs. The hypercube graphs are also unit-distance (1990) showed that every for admits a Hamilton In 1954, Ringel showed that the hypercube graphs admit Hamilton decompositions Hypercube graphs are distance-transitive, Special cases are summarized in the following table. Hypercube graphs may be computed in the Wolfram Language using the command HypercubeGraph,Īnd precomputed properties of hypercube graphs are implemented in the Wolfram Three of the central edges connect to the upper vertex, while the other three connect Space diagonal so that the top and bottom verticesĬoincide, and hence only seven of the cube's eight vertices are visible. The above figures show orthographic projections of some small -hypercube graphs using the first two of each vertex's setĪbove is a projection of the usual cube looking along a The -hypercube graph is also isomorphic to the Hasseĭiagram for the Boolean algebra on elements. The symbols differ in exactly one coordinate. , where or 1 and two vertices are adjacent iff Graph and commonly denoted or, is the graph whose vertices are the symbols.
0 kommentar(er)
