Английская Википедия:Hypercube graph

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Шаблон:Short description Шаблон:Infobox graph; k = 0, \ldots, n\}</math>

| properties = Symmetric
Distance regular
Unit distance
Hamiltonian
Bipartite
| notation = Шаблон:Mvar

}}

In graph theory, the hypercube graph Шаблон:Mvar is the graph formed from the vertices and edges of an Шаблон:Mvar-dimensional hypercube. For instance, the cube graph Шаблон:Math is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Шаблон:Mvar has Шаблон:Math vertices, Шаблон:Math edges, and is a regular graph with Шаблон:Mvar edges touching each vertex.

The hypercube graph Шаблон:Mvar may also be constructed by creating a vertex for each subset of an Шаблон:Mvar-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each Шаблон:Mvar-digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the Шаблон:Mvar-fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of Шаблон:Math connected to each other by a perfect matching.

Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph Шаблон:Mvar that is a cubic graph is the cubical graph Шаблон:Math.

Construction

Файл:Hypercubeconstruction.png

Construction of Шаблон:Math by connecting pairs of corresponding vertices in two copies of Шаблон:Math

The hypercube graph Шаблон:Math may be constructed from the family of subsets of a set with Шаблон:Mvar elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element. Equivalently, it may be constructed using Шаблон:Math vertices labeled with Шаблон:Mvar-bit binary numbers and connecting two vertices by an edge whenever the Hamming distance of their labels is one. These two constructions are closely related: a binary number may be interpreted as a set (the set of positions where it has a Шаблон:Math digit), and two such sets differ in a single element whenever the corresponding two binary numbers have Hamming distance one.

Alternatively, Шаблон:Math may be constructed from the disjoint union of two hypercubes Шаблон:Math, by adding an edge from each vertex in one copy of Шаблон:Math to the corresponding vertex in the other copy, as shown in the figure. The joining edges form a perfect matching.

The above construction gives a recursive algorithm for constructing the adjacency matrix of a hypercube, Шаблон:Math. Copying is done via the Kronecker product, so that the two copies of Шаблон:Math have an adjacency matrix Шаблон:Math ,where Шаблон:Math is the identity matrix in Шаблон:Math dimensions. Meanwhile the joining edges have an adjacency matrix Шаблон:Math. The sum of these two terms gives a recursive function function for the adjacency matrix of a hypercube:

<math>A_{n} = \begin{cases}

1_2\otimes_K A_{n-1}+A_1\otimes_K 1_{2^{n-1}} & \text{if } n>1\\ \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} &\text{if }n=1

\end{cases}</math>

Another construction of Шаблон:Math is the Cartesian product of Шаблон:Mvar two-vertex complete graphs Шаблон:Math. More generally the Cartesian product of copies of a complete graph is called a Hamming graph; the hypercube graphs are examples of Hamming graphs.

Examples

The graph Шаблон:Math consists of a single vertex, while Шаблон:Math is the complete graph on two vertices.

Шаблон:Math is a cycle of length Шаблон:Math.

The graph Шаблон:Math is the 1-skeleton of a cube and is a planar graph with eight vertices and twelve edges.

The graph Шаблон:Math is the Levi graph of the Möbius configuration. It is also the knight's graph for a toroidal <math>4\times 4</math> chessboard.

Properties

Bipartiteness

Every hypercube graph is bipartite: it can be colored with only two colors. The two colors of this coloring may be found from the subset construction of hypercube graphs, by giving one color to the subsets that have an even number of elements and the other color to the subsets with an odd number of elements.

Hamiltonicity

Файл:Gray code tesseract.svg

A Hamiltonian cycle on a tesseract with vertices labelled with a 4-bit cyclic Gray code

Every hypercube Шаблон:Math with Шаблон:Math has a Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path exists between two vertices Шаблон:Mvar and Шаблон:Mvar if and only if they have different colors in a Шаблон:Math-coloring of the graph. Both facts are easy to prove using the principle of induction on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.

Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective correspondence between the set of Шаблон:Mvar-bit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube Шаблон:Math. An analogous property holds for acyclic n-bit Gray codes and Hamiltonian paths.

A lesser known fact is that every perfect matching in the hypercube extends to a Hamiltonian cycle. The question whether every matching extends to a Hamiltonian cycle remains an open problem.

Other properties

The hypercube graph Шаблон:Math (for Шаблон:Math) :

• is the Hasse diagram of a finite Boolean algebra.
• is a median graph. Every median graph is an isometric subgraph of a hypercube, and can be formed as a retraction of a hypercube.
• has more than Шаблон:Math perfect matchings. (this is another consequence that follows easily from the inductive construction.)
• is arc transitive and symmetric. The symmetries of hypercube graphs can be represented as signed permutations.
• contains all the cycles of length Шаблон:Math and is thus a bipancyclic graph.
• can be drawn as a unit distance graph in the Euclidean plane by using the construction of the hypercube graph from subsets of a set of Шаблон:Mvar elements, choosing a distinct unit vector for each set element, and placing the vertex corresponding to the set Шаблон:Mvar at the sum of the vectors in Шаблон:Mvar.
• is a [[k-vertex-connected graph|Шаблон:Mvar-vertex-connected graph]], by Balinski's theorem.
• is planar (can be drawn with no crossings) if and only if Шаблон:Math. For larger values of Шаблон:Mvar, the hypercube has genus Шаблон:Math.
• has exactly <math>2^{2^n-n-1}\prod_{k=2}^n k^Шаблон:N\choose k</math> spanning trees.
• has bandwidth exactly <math>\sum_{i=0}^{n-1} \binom{i}{\lfloor i/2\rfloor}</math>.
• has achromatic number proportional to <math>\sqrt{n2^n}</math>, but the constant of proportionality is not known precisely.
• has as the eigenvalues of its adjacency matrix the numbers Шаблон:Math and as the eigenvalues of its Laplacian matrix the numbers Шаблон:Math. The Шаблон:Mvarth eigenvalue has multiplicity <math>\binom{n}{k}</math> in both cases.
• has isoperimetric number Шаблон:Math.

The family Шаблон:Math for all Шаблон:Math is a Lévy family of graphs.

Problems

Шаблон:Snakes and coils in the box.svg The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snake-in-the-box problem.

Szymanski's conjecture concerns the suitability of a hypercube as a network topology for communications. It states that, no matter how one chooses a permutation connecting each hypercube vertex to another vertex with which it should be connected, there is always a way to connect these pairs of vertices by paths that do not share any directed edge.

See also

• de Bruijn graph
• Cube-connected cycles
• Fibonacci cube
• Folded cube graph
• Frankl–Rödl graph
• Halved cube graph
• Hypercube internetwork topology
• Partial cube

References

• Шаблон:Citation.